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bbscdf


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 -- statistics: P = bbscdf (X, SHAPE, SCALE, LOCATION)

     Birnbaum-Saunders cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Birnbaum-Saunders distribution with parameters
     SHAPE, SCALE, and LOCATION.  The size of P is the common size of X,
     SHAPE, SCALE, and LOCATION.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     See also: bbsinv, bbspdf, bbsrnd.


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Birnbaum-Saunders cumulative distribution function (CDF).



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bbsinv


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 -- statistics: X = bbsinv (P, SHAPE, SCALE, LOCATION)

     Inverse of the Birnbaum-Saunders cumulative distribution function
     (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the Birnbaum-Saunders distribution with parameters
     SHAPE, SCALE, and LOCATION.  The size of X is the common size of P,
     SHAPE, SCALE, and LOCATION.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     See also: bbscdf, bbspdf, bbsrnd.


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Inverse of the Birnbaum-Saunders cumulative distribution function
(iCDF).



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bbspdf


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 -- statistics: Y = bbspdf (X, SHAPE, SCALE, LOCATION)

     Birnbaum-Saunders probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the Birnbaum-Saunders distribution with parameters
     SHAPE, SCALE, and LOCATION.  The size of Y is the common size of X,
     SHAPE, SCALE, and LOCATION.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     See also: bbscdf, bbsinv, bbsrnd.


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Birnbaum-Saunders probability density function (PDF).



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bbsrnd


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 -- statistics: R = bbsrnd (SHAPE, SCALE, LOCATION)
 -- statistics: R = bbsrnd (SHAPE, SCALE, LOCATION, ROWS)
 -- statistics: R = bbsrnd (SHAPE, SCALE, LOCATION, ROWS, COLS, ...)
 -- statistics: R = bbsrnd (SHAPE, SCALE, LOCATION, [SZ])

     Random arrays from the Birnbaum-Saunders distribution.

     'R = bbsrnd (SHAPE, SCALE, LOCATION)' returns an array of random
     numbers chosen from the Birnbaum-Saunders distribution with
     parameters SHAPE, SCALE, and LOCATION.  The size of R is the common
     size of SHAPE, SCALE, and LOCATION.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: bbscdf, bbsinv, bbspdf.


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Random arrays from the Birnbaum-Saunders distribution.



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betacdf


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 -- statistics: P = betacdf (X, A, B)
 -- statistics: P = betacdf (X, A, B, "upper")

     Beta cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     at X of the Beta distribution with parameters A and B.  The size of
     P is the common size of X, A and B.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     'P = betacdf (X, A, B, "upper")' computes the upper tail
     probability of the Beta distribution with parameters A and B at the
     values in X.

     See also: betainv, betapdf, betarnd, betastat.


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Beta cumulative distribution function (CDF).



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betainv


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 -- statistics: X = betainv (P, A, B)

     Inverse of the Beta distribution (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the Beta distribution with parameters A and B.  The
     size of X is the common size of X, A and B.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     See also: betacdf, betapdf, betarnd, betastat.


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Inverse of the Beta distribution (iCDF).



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betapdf


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 -- statistics: Y = betapdf (X, A, B)

     Beta probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the Beta distribution with parameters A and B.  The
     size of Y is the common size of X, A and B.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     See also: betacdf, betainv, betarnd, betastat.


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Beta probability density function (PDF).



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betarnd


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 -- statistics: R = betarnd (A, B)
 -- statistics: R = betarnd (A, B, ROWS)
 -- statistics: R = betarnd (A, B, ROWS, COLS, ...)
 -- statistics: R = betarnd (A, B, [SZ])

     Random arrays from the Beta distribution.

     'R = betarnd (A, B)' returns an array of random numbers chosen from
     the Beta distribution with parameters A and B.  The size of R is
     the common size of A and B.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: betacdf, betainv, betapdf, betastat.


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Random arrays from the Beta distribution.



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binocdf


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 -- statistics: P = binocdf (X, N, PS)
 -- statistics: P = binocdf (X, N, PS, "upper")

     Binomial cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the binomial distribution with parameters N and PS,
     where N is the number of trials and PS is the probability of
     success.  The size of P is the common size of X, N, and PS.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     binocdf (X, N, PS, "upper") computes the complement of the
     cumulative distribution function.

     See also: binoinv, binopdf, binornd, binostat, binotest.


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Binomial cumulative distribution function (CDF).



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binoinv


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 -- statistics: X = binoinv (P, N, PS)

     Inverse of the Binomial cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the binomial distribution with parameters N and PS,
     where N is the number of trials and PS is the probability of
     success.  The size of X is the common size of P, N, and PS.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     See also: binocdf, binopdf, binornd, binostat, binotest.


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Inverse of the Binomial cumulative distribution function (iCDF).



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binopdf


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 -- statistics: Z = binopdf (X, N, PS)

     Binomial probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the binomial distribution with parameters N and PS,
     where N is the number of trials and PS is the probability of
     success.  The size of Y is the common size of X, N, and PS.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     Matlab incompatibility: Octave's 'binopdf' returns NaN for complex
     input values.  Matlab 2021b returns values for complex inputs
     despite the documentation indicates integer and real value inputs
     are required.

     See also: binocdf, binoinv, binornd, binostat, binotest.


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Binomial probability density function (PDF).



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binornd


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 -- statistics: binornd (N, PS)
 -- statistics: binornd (N, PS, ROWS)
 -- statistics: binornd (N, PS, ROWS, COLS, ...)
 -- statistics: binornd (N, PS, [SZ])

     Random arrays from the Binomial distribution

     Return a matrix of random samples from the binomial distribution
     with parameters N and PS, where N is the number of trials and PS is
     the probability of success.  The size of R is the common size of N
     and PS.  A scalar input functions as a constant matrix of the same
     size as the other inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: binocdf, binoinv, binopdf, binostat, binotest.


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Random arrays from the Binomial distribution



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burrcdf


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 -- statistics: P = burrcdf (X, C, K)

     Burr type XII cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Burr type XII distribution with parameters A, C,
     and K.  The size of X is the common size of P, A, C, and K.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     See also: burrinv, burrpdf, burrrnd.


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Burr type XII cumulative distribution function (CDF).



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burrinv


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 -- statistics: X = burrinv (P, A, C, K)

     Inverse of the Burr type XII cumulative distribution function
     (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the Burr type XII distribution with parameters A, C,
     and K.  The size of X is the common size of P, A, C, and K.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     See also: burrcdf, burrpdf, burrrnd.


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Inverse of the Burr type XII cumulative distribution function (iCDF).



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burrpdf


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 -- statistics: Y = burrpdf (X, A, C, K)

     Burr type XII probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the Burr type XII distribution with parameters A, C,
     and K.  The size of Y is the common size of X, A, C, and K.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     See also: burrcdf, burrinv, burrrnd.


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Burr type XII probability density function (PDF).



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burrrnd


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 -- statistics: R = burrrnd (A, C, K)
 -- statistics: R = burrrnd (A, C, K, ROWS)
 -- statistics: R = burrrnd (A, C, K, ROWS, COLS, ...)
 -- statistics: R = burrrnd (A, C, K, [SZ])

     Random arrays from the Burr type XII distribution.

     'R = burrrnd (A, C, K)' returns an array of random numbers chosen
     from the Burr type XII distribution with parameters A, C, and K.
     The size of R is the common size of A, C, and K.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: burrcdf, burrinv, burrpdf.


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Random arrays from the Burr type XII distribution.



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bvncdf


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 -- statistics: P = bvncdf (X, MU, SIGMA)
 -- statistics: P = bvncdf (X, [], SIGMA)

     Bivariate normal cumulative distribution function (CDF).

     'P = bvncdf (X, MU, SIGMA)' will compute the bivariate normal
     cumulative distribution function of X given a mean MU, which must
     be a scalar, and a 2x2 SIGMA covariance matrix, which must be
     positive definite.

     See also: mvncdf.


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Bivariate normal cumulative distribution function (CDF).



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cauchy_cdf


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 -- statistics: P = cauchy_cdf (X)
 -- statistics: P = cauchy_cdf (X, LOCATION)
 -- statistics: P = cauchy_cdf (X, LOCATION, SCALE)

     Cauchy cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Cauchy distribution with location parameter
     LOCATION and scale parameter SCALE.  The size of P is the common
     size of X, LOCATION, and SCALE.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     Default values are LOCATION = 0, SCALE = 1.

     See also: cauchy_inv, cauchy_pdf, cauchy_rnd.


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Cauchy cumulative distribution function (CDF).



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cauchy_inv


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 -- statistics: X = cauchy_inv (P)
 -- statistics: X = cauchy_inv (P, LOCATION, SCALE)

     Inverse of the Cauchy cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the Cauchy distribution with location parameter
     LOCATION and scale parameter SCALE.  The size of X is the common
     size of P, LOCATION, and SCALE.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     Default values are LOCATION = 0, SCALE = 1.

     See also: cauchy_cdf, cauchy_pdf, cauchy_rnd.


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Inverse of the Cauchy cumulative distribution function (iCDF).



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cauchy_pdf


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 -- statistics: Y = cauchy_pdf (X)
 -- statistics: Y = cauchy_pdf (X, LOCATION, SCALE)

     Cauchy probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the Cauchy distribution with location parameter
     LOCATION and scale parameter SCALE.  The size of Y is the common
     size of X, LOCATION, and SCALE.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     Default values are LOCATION = 0, SCALE = 1.

     See also: cauchy_cdf, cauchy_inv, cauchy_rnd.


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Cauchy probability density function (PDF).



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cauchy_rnd


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 -- statistics: R = cauchy_rnd (LOCATION, SCALE)
 -- statistics: R = cauchy_rnd (LOCATION, SCALE, ROWS)
 -- statistics: R = cauchy_rnd (LOCATION, SCALE, ROWS, COLS, ...)
 -- statistics: R = cauchy_rnd (LOCATION, SCALE, [SZ])

     Random arrays from the Cauchy distribution.

     'R = cauchy_rnd (LOCATION, SCALE)' returns an array of random
     numbers chosen from the Cauchy distribution with parameters
     LOCATION and SCALE.  The size of R is the common size of LOCATION
     and SCALE.  A scalar input functions as a constant matrix of the
     same size as the other inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: cauchy_cdf, cauchy_inv, cauchy_pdf.


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Random arrays from the Cauchy distribution.



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chi2cdf


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 -- statistics: P = tcdf (X, DF)
 -- statistics: P = tcdf (X, DF, "upper")

     Chi-square cumulative distribution function.

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Chi-Squared distribution with DF degrees of
     freedom.  The chi-square density function with DF degrees of
     freedom is the same as a gamma density function with parameters
     DF/2 and 2.

     The size of P is the common size of X and DF.  A scalar input
     functions as a constant matrix of the same size as the other input.

     'P = fcdf (X, DF, "upper")' computes the upper tail probability of
     the Chi-Squared distribution with DF degrees of freedom at the
     values in X.

     See also: chi2inv, chi2pdf, chi2rnd, chi2stat.


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Chi-square cumulative distribution function.



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chi2inv


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 -- statistics: X = chi2inv (P, DF)

     Inverse of the Chi-square cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the chi-square distribution with DF degrees of
     freedom.  The size of X is the common size of P and DF.  A scalar
     input functions as a constant matrix of the same size as the other
     inputs.

     See also: chi2cdf, chi2pdf, chi2rnd, chi2stat.


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Inverse of the Chi-square cumulative distribution function (iCDF).



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chi2pdf


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 -- statistics: Y = chi2pdf (X, DF)

     Chi-square probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the chi-square distribution with DF degrees of
     freedom.  The size of Y is the common size of X and DF.  A scalar
     input functions as a constant matrix of the same size as the other
     inputs.

     See also: chi2cdf, chi2inv, chi2rnd, chi2stat.


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Chi-square probability density function (PDF).



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chi2rnd


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 -- statistics: R = chi2rnd (DF)
 -- statistics: R = chi2rnd (DF, ROWS)
 -- statistics: R = chi2rnd (DF, ROWS, COLS, ...)
 -- statistics: R = chi2rnd (DF, [SZ])

     Random arrays from the Chi-square distribution.

     'R = chi2rnd (DF)' returns an array of random numbers chosen from
     the Chi-square distribution with DF degrees of freedom.  The size
     of R is the size of DF.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: chi2cdf, chi2inv, chi2pdf, chi2stat.


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Random arrays from the Chi-square distribution.



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copulacdf


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 -- statistics: P = copulacdf (FAMILY, X, THETA)
 -- statistics: P = copulacdf ('t', X, THETA, DF)

     Copula family cumulative distribution functions (CDF).

     Arguments
     ---------

        * FAMILY is the copula family name.  Currently, FAMILY can be
          ''Gaussian'' for the Gaussian family, ''t'' for the Student's
          t family, ''Clayton'' for the Clayton family, ''Gumbel'' for
          the Gumbel-Hougaard family, ''Frank'' for the Frank family,
          ''AMH'' for the Ali-Mikhail-Haq family, or ''FGM'' for the
          Farlie-Gumbel-Morgenstern family.

        * X is the support where each row corresponds to an observation.

        * THETA is the parameter of the copula.  For the Gaussian and
          Student's t copula, THETA must be a correlation matrix.  For
          bivariate copulas THETA can also be a correlation coefficient.
          For the Clayton family, the Gumbel-Hougaard family, the Frank
          family, and the Ali-Mikhail-Haq family, THETA must be a vector
          with the same number of elements as observations in X or be
          scalar.  For the Farlie-Gumbel-Morgenstern family, THETA must
          be a matrix of coefficients for the Farlie-Gumbel-Morgenstern
          polynomial where each row corresponds to one set of
          coefficients for an observation in X.  A single row is
          expanded.  The coefficients are in binary order.

        * DF is the degrees of freedom for the Student's t family.  DF
          must be a vector with the same number of elements as
          observations in X or be scalar.

     Return values
     -------------

        * P is the cumulative distribution of the copula at each row of
          X and corresponding parameter THETA.

     Examples
     --------

          x = [0.2:0.2:0.6; 0.2:0.2:0.6];
          theta = [1; 2];
          p = copulacdf ("Clayton", x, theta)

          x = [0.2:0.2:0.6; 0.2:0.1:0.4];
          theta = [0.2, 0.1, 0.1, 0.05];
          p = copulacdf ("FGM", x, theta)

     References
     ----------

       1. Roger B. Nelsen.  'An Introduction to Copulas'.  Springer, New
          York, second edition, 2006.

     See also: copulapdf, copularnd.


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Copula family cumulative distribution functions (CDF).



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copulapdf


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# length: 1520
 -- statistics: Y = copulapdf (FAMILY, X, THETA)

     Copula family probability density functions (PDF).

     Arguments
     ---------

        * FAMILY is the copula family name.  Currently, FAMILY can be
          ''Clayton'' for the Clayton family, ''Gumbel'' for the
          Gumbel-Hougaard family, ''Frank'' for the Frank family, or
          ''AMH'' for the Ali-Mikhail-Haq family.

        * X is the support where each row corresponds to an observation.

        * THETA is the parameter of the copula.  The elements of THETA
          must be greater than or equal to '-1' for the Clayton family,
          greater than or equal to '1' for the Gumbel-Hougaard family,
          arbitrary for the Frank family, and greater than or equal to
          '-1' and lower than '1' for the Ali-Mikhail-Haq family.
          Moreover, THETA must be non-negative for dimensions greater
          than '2'.  THETA must be a column vector with the same number
          of rows as X or be scalar.

     Return values
     -------------

        * Y is the probability density of the copula at each row of X
          and corresponding parameter THETA.

     Examples
     --------

          x = [0.2:0.2:0.6; 0.2:0.2:0.6];
          theta = [1; 2];
          y = copulapdf ("Clayton", x, theta)

          y = copulapdf ("Gumbel", x, 2)

     References
     ----------

       1. Roger B. Nelsen.  'An Introduction to Copulas'.  Springer, New
          York, second edition, 2006.

     See also: copulacdf, copularnd.


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Copula family probability density functions (PDF).



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copularnd


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# elements: 1
# length: 1901
 -- Function File: R = copularnd (FAMILY, THETA, N)
 -- Function File: R = copularnd (FAMILY, THETA, N, D)
 -- Function File: R = copularnd ('t', THETA, DF, N)

     Random arrays from the copula family distributions.

     Arguments
     ---------

        * FAMILY is the copula family name.  Currently, FAMILY can be
          ''Gaussian'' for the Gaussian family, ''t'' for the Student's
          t family, or ''Clayton'' for the Clayton family.

        * THETA is the parameter of the copula.  For the Gaussian and
          Student's t copula, THETA must be a correlation matrix.  For
          bivariate copulas THETA can also be a correlation coefficient.
          For the Clayton family, THETA must be a vector with the same
          number of elements as samples to be generated or be scalar.

        * DF is the degrees of freedom for the Student's t family.  DF
          must be a vector with the same number of elements as samples
          to be generated or be scalar.

        * N is the number of rows of the matrix to be generated.  N must
          be a non-negative integer and corresponds to the number of
          samples to be generated.

        * D is the number of columns of the matrix to be generated.  D
          must be a positive integer and corresponds to the dimension of
          the copula.

     Return values
     -------------

        * R is a matrix of random samples from the copula with N samples
          of distribution dimension D.

     Examples
     --------

          theta = 0.5;
          r = copularnd ("Gaussian", theta);

          theta = 0.5;
          df = 2;
          r = copularnd ("t", theta, df);

          theta = 0.5;
          n = 2;
          r = copularnd ("Clayton", theta, n);

     References
     ----------

       1. Roger B. Nelsen.  'An Introduction to Copulas'.  Springer, New
          York, second edition, 2006.


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Random arrays from the copula family distributions.



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evcdf


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# length: 1785
 -- statistics: P = evcdf (X)
 -- statistics: P = evcdf (X, MU)
 -- statistics: P = evcdf (X, MU, SIGMA)
 -- statistics: P = evcdf (..., "upper")
 -- statistics: [P, PLO, PUP] = evcdf (X, MU, SIGMA, PCOV)
 -- statistics: [P, PLO, PUP] = evcdf (X, MU, SIGMA, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = evcdf (..., "upper")

     Extreme value cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) of the type 1 extreme values CDF at X of the normal
     distribution with location parameter MU and scale parameter SIGMA.
     The size of P is the common size of X, MU and SIGMA.  A scalar
     input functions as a constant matrix of the same size as the other
     inputs.

     Default values are MU = 0, SIGMA = 1.

     When called with three output arguments, '[P, PLO, PUP]' it
     computes the confidence bounds for P when the input parameters MU
     and SIGMA are estimates.  In such case, PCOV, a 2-by-2 matrix
     containing the covariance matrix of the estimated parameters, is
     necessary.  ALPHA has a default value of 0.05, and specifies 100 *
     (1 - ALPHA)% confidence bounds.  PLO and PUP are arrays of the same
     size as P containing the lower and upper confidence bounds.

     The type 1 extreme value distribution is also known as the Gumbel
     distribution.  The version used here is suitable for modeling
     minima; the mirror image of this distribution can be used to model
     maxima by negating X.  If Y has a Weibull distribution, then 'X =
     log (Y)' has the type 1 extreme value distribution.

     '[...] = evcdf (..., "upper")' computes the upper tail probability
     of the extreme value distribution.

     See also: evinv, evpdf, evrnd, evfit, evlike, evstat.


# name: <cell-element>
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Extreme value cumulative distribution function (CDF).



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# length: 5
evinv


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# length: 1560
 -- statistics: X = evinv (P)
 -- statistics: X = evcdf (P, MU)
 -- statistics: X = evinv (P, MU, SIGMA)
 -- statistics: [X, XLO, XUP] = evinv (X, MU, SIGMA, PCOV)
 -- statistics: [X, XLO, XUP] = evinv (X, MU, SIGMA, PCOV, ALPHA)

     Inverse of the extreme value cumulative distribution function
     (iCDF).

     For each element of P, compute the inverse cdf for a type 1 extreme
     value distribution with location parameter MU and scale parameter
     SIGMA.  The size of X is the common size of P, MU and SIGMA.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     Default values are MU = 0, SIGMA = 1.

     When called with three output arguments, '[X, XLO, XUP]' it
     computes the confidence bounds for X when the input parameters MU
     and SIGMA are estimates.  In such case, PCOV, a 2-by-2 matrix
     containing the covariance matrix of the estimated parameters, is
     necessary.  Optionally, ALPHA has a default value of 0.05, and
     specifies 100 * (1 - ALPHA)% confidence bounds.  XLO and XUP are
     arrays of the same size as X containing the lower and upper
     confidence bounds.

     The type 1 extreme value distribution is also known as the Gumbel
     distribution.  The version used here is suitable for modeling
     minima; the mirror image of this distribution can be used to model
     maxima by negating X.  If Y has a Weibull distribution, then 'X =
     log (Y)' has the type 1 extreme value distribution.

     See also: evcdf, evpdf, evrnd, evfit, evlike, evstat.


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Inverse of the extreme value cumulative distribution function (iCDF).



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# type: sq_string
# elements: 1
# length: 5
evpdf


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# length: 906
 -- statistics: Y = evpdf (X)
 -- statistics: Y = evpdf (X, MU)
 -- statistics: Y = evpdf (X, MU, SIGMA)

     Extreme value probability density function (PDF).

     'Y = evpdf (X, MU, SIGMA)' returns the pdf of the type 1 extreme
     value distribution with location parameter MU and scale parameter
     SIGMA.  The size of X is the common size of P, MU and SIGMA.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     Default values are MU = 0, SIGMA = 1.

     The type 1 extreme value distribution is also known as the Gumbel
     distribution.  The version used here is suitable for modeling
     minima; the mirror image of this distribution can be used to model
     maxima by negating X.  If Y has a Weibull distribution, then 'X =
     log (Y)' has the type 1 extreme value distribution.

     See also: evcdf, evinv, evrnd, evfit, evlike, evstat.


# name: <cell-element>
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Extreme value probability density function (PDF).



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# elements: 1
# length: 5
evrnd


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# length: 1037
 -- statistics: R = evrnd (MU, SIGMA)
 -- statistics: R = evrnd (MU, SIGMA, M, N, ...)
 -- statistics: R = evrnd (MU, SIGMA, [M, N, ...])

     Random arrays from the extreme value distribution.

     'R = evrnd (MU, SIGMA)' returns an array of random numbers chosen
     from the type 1 extreme value distribution with location parameter
     MU and scale parameter SIGMA.  The size of R is the common size of
     MU and SIGMA.  A scalar input functions as a constant matrix of the
     same size as the other inputs.

     'R = evrnd (MU, SIGMA, M, N, ...)' or 'R = evrnd (MU, SIGMA, [M, N,
     ...])' returns an M-by-N-by-...  array.

     The type 1 extreme value distribution is also known as the Gumbel
     distribution.  The version used here is suitable for modeling
     minima; the mirror image of this distribution can be used to model
     maxima by negating X.  If Y has a Weibull distribution, then 'X =
     log (Y)' has the type 1 extreme value distribution.

     See also: evcdf, evinv, evpdf, evfit, evlike, evstat.


# name: <cell-element>
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# length: 50
Random arrays from the extreme value distribution.



# name: <cell-element>
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# length: 6
expcdf


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# type: sq_string
# elements: 1
# length: 1299
 -- statistics: P = expcdf (X)
 -- statistics: P = expcdf (X, MU)
 -- statistics: P = expcdf (..., "upper")
 -- statistics: [P, PLO, PUP] = expcdf (X, MU, PCOV)
 -- statistics: [P, PLO, PUP] = expcdf (X, MU, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = expcdf (..., "upper")

     Exponential cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the exponential distribution with mean MU.  The size
     of P is the common size of X, MU and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     Default value for MU = 1.

     The arguments can be of common size or scalars.

     When called with three output arguments, '[P, PLO, PUP]' it
     computes the confidence bounds for P when the input parameter MU is
     an estimate.  In such case, PCOV is the variance of the estimated
     MU.  ALPHA has a default value of 0.05, and specifies 100 * (1 -
     ALPHA)% confidence bounds.  PLO and PUP are arrays of the same size
     as P containing the lower and upper confidence bounds.

     '[...] = expcdf (..., "upper")' computes the upper tail probability
     of the exponential distribution.

     See also: expinv, exppdf, exprnd, expfit, explike, expstat.


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Exponential cumulative distribution function (CDF).



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# length: 6
expinv


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# length: 447
 -- statistics: X = expinv (P, MU)

     Inverse of the exponential cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the exponential distribution with mean MU.  The size
     of X is the common size of P and MU.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     See also: expcdf, exppdf, exprnd, expfit, explike, expstat.


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Inverse of the exponential cumulative distribution function (iCDF).



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# type: sq_string
# elements: 1
# length: 6
exppdf


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# length: 428
 -- statistics: Y = exppdf (X, MU)

     Exponential probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the exponential distribution with mean MU.  The size
     of Y is the common size of X and MU.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     See also: expcdf, expinv, exprnd, expfit, explike, expstat.


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Exponential probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
exprnd


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# type: sq_string
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# length: 814
 -- statistics: R = exprnd (MU)
 -- statistics: R = exprnd (MU, ROWS)
 -- statistics: R = exprnd (MU, ROWS, COLS, ...)
 -- statistics: R = exprnd (MU, [SZ])

     Random arrays from the exponential distribution.

     'R = exprnd (MU)' returns an array of random numbers chosen from
     the exponential distribution with mean parameter MU.  The size of R
     is the size of MU.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: expcdf, expinv, exppdf, expfit, explike, expstat.


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Random arrays from the exponential distribution.



# name: <cell-element>
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# elements: 1
# length: 4
fcdf


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# length: 647
 -- statistics: P = tcdf (X, DF1, DF2)
 -- statistics: P = tcdf (X, DF1, DF2, "upper")

     F cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the F distribution with DF1 and DF2 degrees of
     freedom.

     The size of P is the common size of X, DF1, and DF2.  A scalar
     input functions as a constant matrix of the same size as the other
     inputs.

     'P = fcdf (X, DF1, DF2, "upper")' computes the upper tail
     probability of the F distribution with DF1 and DF2 degrees of
     freedom at the values in X.

     See also: finv, fpdf, frnd, fstat.


# name: <cell-element>
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F cumulative distribution function (CDF).



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# elements: 1
# length: 4
finv


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# type: sq_string
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# length: 441
 -- statistics: X = finv (P, DF1, DF2)

     Inverse of the F cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the F distribution with DF1 and DF2 degrees of
     freedom.  The size of X is the common size of P, DF1, and DF2.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     See also: fcdf, fpdf, frnd, fstat.


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Inverse of the F cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
fpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 422
 -- statistics: Y = fpdf (X, DF1, DF2)

     F probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the F distribution with DF1 and DF2 degrees of
     freedom.  The size of Y is the common size of X, DF1, and DF2.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     See also: fcdf, finv, frnd, fstat.


# name: <cell-element>
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# length: 37
F probability density function (PDF).



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# type: sq_string
# elements: 1
# length: 4
frnd


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# length: 913
 -- statistics: R = frnd (DF1, DF2)
 -- statistics: R = frnd (DF1, DF2, ROWS)
 -- statistics: R = frnd (DF1, DF2, ROWS, COLS, ...)
 -- statistics: R = frnd (DF1, DF2, [SZ])

     Random arrays from the F distribution.

     'R = frnd (DF1, DF2)' returns an array of random numbers chosen
     from the F distribution with DF1 and DF2 degrees of freedom.  The
     size of R is the common size of DF1 and DF2.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: fcdf, finv, fpdf, fstat.


# name: <cell-element>
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# length: 38
Random arrays from the F distribution.



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# type: sq_string
# elements: 1
# length: 6
gamcdf


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# type: sq_string
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# length: 1367
 -- statistics: P = gamcdf (X, A)
 -- statistics: P = gamcdf (X, A, B)
 -- statistics: P = gamcdf (..., "upper")
 -- statistics: [P, PLO, PUP] = evcdf (X, A, B, PCOV)
 -- statistics: [P, PLO, PUP] = evcdf (X, A, B, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = evcdf (..., "upper")

     Gamma cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Gamma distribution with shape parameter A and
     scale B.  When called with only one parameter, then B defaults to
     1.  The size of P is the common size of X, A, and B.  A scalar
     input functions as a constant matrix of the same size as the other
     inputs.

     When called with three output arguments, '[P, PLO, PUP]' it
     computes the confidence bounds for P when the input parameters A
     and B are estimates.  In such case, PCOV, a 2-by-2 matrix
     containing the covariance matrix of the estimated parameters, is
     necessary.  Optionally, ALPHA has a default value of 0.05, and
     specifies 100 * (1 - ALPHA)% confidence bounds.  PLO and PUP are
     arrays of the same size as P containing the lower and upper
     confidence bounds.

     '[...] = gamcdf (..., "upper")' computes the upper tail probability
     of the gamma distribution.

     See also: gaminv, gampdf, gamrnd, gamfit, gamlike, gamstat.


# name: <cell-element>
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Gamma cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gaminv


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# type: sq_string
# elements: 1
# length: 467
 -- statistics: X = gaminv (P, A, B)

     Inverse of the Gamma cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the Gamma distribution with shape parameter A and
     scale B.  The size of X is the common size of P, A, and B.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     See also: gamcdf, gampdf, gamrnd, gamfit, gamlike, gamstat.


# name: <cell-element>
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# length: 61
Inverse of the Gamma cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gampdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 456
 -- statistics: Y = gampdf (X, A, B)

     Gamma probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the Gamma distribution with shape parameter A and
     scale parameter B.  The size of Y is the common size of X, A and B.
     A scalar input functions as a constant matrix of the same size as
     the other inputs.

     See also: gamcdf, gaminv, gamrnd, gamfit, gamlike, gamstat.


# name: <cell-element>
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# length: 41
Gamma probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gamrnd


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# length: 941
 -- statistics: R = gamrnd (A, B)
 -- statistics: R = gamrnd (A, B, ROWS)
 -- statistics: R = gamrnd (A, B, ROWS, COLS, ...)
 -- statistics: R = gamrnd (A, B, [SZ])

     Random arrays from the Gamma distribution.

     'R = gamrnd (A, B)' returns an array of random numbers chosen from
     the Gamma distribution with shape parameter A and scale parameter
     B.  The size of R is the common size of A and B.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: gamcdf, gaminv, gampdf, gamfit, gamlike, gamstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 42
Random arrays from the Gamma distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
geocdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 741
 -- statistics: P = geocdf (X, PS)
 -- statistics: P = geocdf (X, PS, "upper")

     Geometric cumulative distribution function (CDF).

     The geometric distribution models the number of failures (X) of a
     Bernoulli trial with probability PS before the first success.

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the geometric distribution with parameter PS.  The
     size of P is the common size of X and PS.  A scalar input functions
     as a constant matrix of the same size as the other inputs.

     'P = geocdf (X, PS, "upper")' computes the upper tail probability
     of the geometric distribution with probability parameter PS.

     See also: geoinv, geopdf, geornd, geostat.


# name: <cell-element>
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# elements: 1
# length: 49
Geometric cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
geoinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 570
 -- statistics: X = geoinv (P, PS)

     Inverse of the geometric cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the geometric distribution with parameter PS.  The
     size of X is the common size of P and PS.  A scalar input functions
     as a constant matrix of the same size as the other inputs.

     The geometric distribution models the number of failures (P) of a
     Bernoulli trial with probability PS before the first success.

     See also: geocdf, geopdf, geornd, geostat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 65
Inverse of the geometric cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
geopdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 551
 -- statistics: Y = geopdf (X, PS)

     Geometric probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the geometric distribution with parameter PS.  The
     size of Y is the common size of X and PS.  A scalar input functions
     as a constant matrix of the same size as the other inputs.

     The geometric distribution models the number of failures (X) of a
     Bernoulli trial with probability PS before the first success.

     See also: geocdf, geoinv, geornd, geostat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 45
Geometric probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
geornd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 935
 -- statistics: R = geornd (PS)
 -- statistics: R = geornd (PS, ROWS)
 -- statistics: R = geornd (PS, ROWS, COLS, ...)
 -- statistics: R = geornd (PS, [SZ])

     Random arrays from the geometric distribution.

     'R = geornd (PS)' returns an array of random numbers chosen from
     the Birnbaum-Saunders distribution with parameter PS.  The size of
     R is the size of PS.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     The geometric distribution models the number of failures (X) of a
     Bernoulli trial with probability PS before the first success.

     See also: geocdf, geoinv, geopdf, geostat.


# name: <cell-element>
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Random arrays from the geometric distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1902
 -- statistics: P = gevcdf (X)
 -- statistics: P = gevcdf (X, K)
 -- statistics: P = gevcdf (X, K, SIGMA)
 -- statistics: P = gevcdf (X, K, SIGMA, MU)
 -- statistics: P = gevcdf (..., "upper")

     Generalized extreme value (GEV) cumulative distribution function
     (CDF).

     'P = gevcdf (X, K, SIGMA, MU)' returns the CDF of the generalized
     extreme value (GEV) distribution with shape parameter K, scale
     parameter SIGMA, and location parameter MU, evaluated at the values
     in X.  The size of P is the common size of the input arguments.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     Default values for K, SIGMA, and MU are 0, 1, and 0, respectively.

     When K < 0, the GEV is the type III extreme value distribution.
     When K > 0, the GEV distribution is the type II, or Frechet,
     extreme value distribution.  If W has a Weibull distribution as
     computed by the 'wblcdf' function, then -W has a type III extreme
     value distribution and 1/W has a type II extreme value
     distribution.  In the limit as K approaches 0, the GEV is the
     mirror image of the type I extreme value distribution as computed
     by the 'evcdf' function.

     The mean of the GEV distribution is not finite when K >= 1, and the
     variance is not finite when K >= 1/2.  The GEV distribution has
     positive density only for values of X such that K*(X-MU)/SIGMA >
     -1.

     'P = gevcdf (..., "upper")' returns the upper tail probability of
     the generalized extreme value distribution.

     References
     ----------

       1. Rolf-Dieter Reiss and Michael Thomas.  'Statistical Analysis
          of Extreme Values with Applications to Insurance, Finance,
          Hydrology and Other Fields'.  Chapter 1, pages 16-17,
          Springer, 2007.

     See also: gevinv, gevpdf, gevrnd, gevfit, gevlike, gevstat.


# name: <cell-element>
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Generalized extreme value (GEV) cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 887
 -- statistics: X = gevinv (P, K, SIGMA, MU)

     Inverse of the generalized extreme value (GEV) cumulative
     distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the GEV distribution with shape parameter K, scale
     parameter SIGMA, and location parameter MU.  The size of X is the
     common size of the input arguments.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     Default values for K, SIGMA, and MU are 0, 1, and 0, respectively.

     References
     ----------

       1. Rolf-Dieter Reiss and Michael Thomas.  'Statistical Analysis
          of Extreme Values with Applications to Insurance, Finance,
          Hydrology and Other Fields'.  Chapter 1, pages 16-17,
          Springer, 2007.

     See also: gevcdf, gevpdf, gevrnd, gevfit, gevlike, gevstat.


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Inverse of the generalized extreme value (GEV) cumulative distribution
functi...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 790
 -- statistics: Y = gevpdf (X, K, SIGMA, MU)

     Generalized extreme value (GEV) probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the GEV distribution with shape parameter K, scale
     parameter SIGMA, and location parameter MU.  The size of X is the
     common size of the input arguments.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     References
     ----------

       1. Rolf-Dieter Reiss and Michael Thomas.  'Statistical Analysis
          of Extreme Values with Applications to Insurance, Finance,
          Hydrology and Other Fields'.  Chapter 1, pages 16-17,
          Springer, 2007.

     See also: gevcdf, gevinv, gevrnd, gevfit, gevlike, gevstat.


# name: <cell-element>
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Generalized extreme value (GEV) probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1298
 -- statistics: gevrnd (K, SIGMA, MU)
 -- statistics: gevrnd (K, SIGMA, MU, ROWS)
 -- statistics: gevrnd (K, SIGMA, MU, ROWS, COLS, ...)
 -- statistics: gevrnd (K, SIGMA, MU, [SZ])

     Random arrays from the generalized extreme value (GEV)
     distribution.

     'R = gevrnd (K, SIGMA, MU' returns an array of random numbers
     chosen from the GEV distribution with shape parameter K, scale
     parameter SIGMA, and location parameter MU.  The size of R is the
     common size of the input arguments.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     When called with a single size argument, returns a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector SZ
     of dimensions.

     References
     ----------

       1. Rolf-Dieter Reiss and Michael Thomas.  'Statistical Analysis
          of Extreme Values with Applications to Insurance, Finance,
          Hydrology and Other Fields'.  Chapter 1, pages 16-17,
          Springer, 2007.

     See also: gevcdf, gevinv, gevpdf, gevfit, gevlike, gevstat.


# name: <cell-element>
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Random arrays from the generalized extreme value (GEV) distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
gpcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1385
 -- statistics: P = gpcdf (X)
 -- statistics: P = gpcdf (X, SHAPE)
 -- statistics: P = gpcdf (X, SHAPE, SCALE)
 -- statistics: P = gpcdf (X, SHAPE, SCALE, LOCATION)
 -- statistics: P = gpcdf (..., "upper")

     Generalized Pareto cumulative distribution function (cdf).

     Compute the cumulative distribution function (CDF) at X of the
     generalized Pareto distribution with parameters LOCATION, SCALE,
     and SHAPE.  The size of P is the common size of the input
     arguments.  A scalar input functions as a constant matrix of the
     same size as the other inputs.

     Default values for SHAPE, SCALE, and LOCATION are 0, 1, and 0,
     respectively.

     When 'SHAPE = 0' and 'LOCATION = 0', the Generalized Pareto CDF is
     equivalent to the exponential distribution.  When 'SHAPE > 0' and
     'LOCATION = SHAPE / SHAPE' the Generalized Pareto is equivalent to
     the Pareto distribution.  The mean of the Generalized Pareto is not
     finite when 'SHAPE >= 1' and the variance is not finite when 'SHAPE
     >= 1/2'.  When 'SHAPE >= 0', the Generalized Pareto has positive
     density for 'X > LOCATION', or, when 'LOCATION < 0',for '0 <= (X -
     LOCATION) / SCALE <= -1 / SHAPE'.

     '[...] = gpcdf(..., "upper")' returns the upper tail probability of
     the generalized Pareto distribution.

     See also: gpinv, gppdf, gprnd, gpfit, gplike, gpstat.


# name: <cell-element>
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Generalized Pareto cumulative distribution function (cdf).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
gpinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 525
 -- statistics: X = gpinv (P, SHAPE, SCALE, LOCATION)

     Inverse of the generalized Pareto cumulative distribution function
     (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the generalized Pareto distribution with parameters
     LOCATION, SCALE, and SHAPE.  The size of X is the common size of
     the input arguments.  A scalar input functions as a constant matrix
     of the same size as the other inputs.

     See also: gpcdf, gppdf, gprnd, gpfit, gplike, gpstat.


# name: <cell-element>
# type: sq_string
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# length: 74
Inverse of the generalized Pareto cumulative distribution function
(iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
gppdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 511
 -- statistics: Y = gppdf (X, SHAPE, SCALE, LOCATION)

     Generalized Pareto probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the generalized Pareto distribution with parameters
     SHAPE, SCALE, and LOCATION.  The size of Y is the common size of X,
     SHAPE, SCALE, and LOCATION.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     See also: gpcdf, gpinv, gprnd, gpfit, gplike, gpstat.


# name: <cell-element>
# type: sq_string
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Generalized Pareto probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
gprnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1063
 -- statistics: R = gprnd (SHAPE, SCALE, LOCATION)
 -- statistics: R = gprnd (SHAPE, SCALE, LOCATION, ROWS)
 -- statistics: R = gprnd (SHAPE, SCALE, LOCATION, ROWS, COLS, ...)
 -- statistics: R = gprnd (SHAPE, SCALE, LOCATION, [SZ])

     Random arrays from the generalized Pareto distribution.

     'R = gprnd (SHAPE, SCALE, LOCATION)' returns an array of random
     numbers chosen from the generalized Pareto distribution with
     parameters SHAPE, SCALE, and LOCATION.  The size of R is the common
     size of SHAPE, SCALE, and LOCATION.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: gpcdf, gpinv, gppdf, gpfit, gplike, gpstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 55
Random arrays from the generalized Pareto distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
hygecdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 817
 -- statistics: P = hygecdf (X, T, M, N)
 -- statistics: P = hygecdf (X, T, M, N, "upper")

     Hypergeometric cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the hypergeometric distribution with parameters T, K,
     and N.  The size of P is the common size of the input arguments.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     This is the probability of obtaining not more than X marked items
     when randomly drawing a sample of size N without replacement from a
     population of total size T containing M marked items.

     The parameters T, M, and N must be positive integers with M and N
     not greater than T.

     See also: hygeinv, hygepdf, hygernd, hygestat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 54
Hypergeometric cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
hygeinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 764
 -- statistics: P = hygeinv (X, T, M, N)

     Inverse of the hypergeometric cumulative distribution function
     (iCDF).

     For each element of X, compute the quantile (the inverse of the
     CDF) at X of the hypergeometric distribution with parameters T, M,
     and N.  The size of P is the common size of the input parameters.
     A scalar input functions as a constant matrix of the same size as
     the other inputs.

     This is the probability of obtaining X marked items when randomly
     drawing a sample of size N without replacement from a population of
     total size T containing M marked items.  The parameters T, M, and N
     must be positive integers with M and N not greater than T.

     See also: hygeinv, hygepdf, hygernd, hygestat.


# name: <cell-element>
# type: sq_string
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Inverse of the hypergeometric cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
hygepdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1184
 -- statistics: Y = hygepdf (X, T, M, N)
 -- statistics: Y = hygepdf (..., "vectorexpand")

     Hypergeometric probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the hypergeometric distribution with parameters T, M,
     and N.  The size of Y is the common size of the input parameters.
     A scalar input functions as a constant matrix of the same size as
     the other inputs.

     This is the probability of obtaining X marked items when randomly
     drawing a sample of size N without replacement from a population of
     total size T containing M marked items.  The parameters T, M, and N
     must be positive integers with M and N not greater than T.

     If the optional parameter 'vectorexpand' is provided, X may be an
     array with size different from parameters T, M, and N (which must
     still be of a common size or scalar).  Each element of X will be
     evaluated against each set of parameters T, M, and N in columnwise
     order.  The output Y will be an array of size 'R x S', where 'R =
     numel (T)', and 'S = numel (X)'.

     See also: hygecdf, hygeinv, hygernd, hygestat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 50
Hypergeometric probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
hygernd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1042
 -- statistics: hygernd (T, M, N)
 -- statistics: hygernd (T, M, N, ROWS)
 -- statistics: hygernd (T, M, N, ROWS, COLS, ...)
 -- statistics: hygernd (T, M, N, [SZ])

     Random arrays from the hypergeometric distribution.

     'R = hygernd ((T, M, N' returns an array of random numbers chosen
     from the hypergeometric distribution with parameters T, M, and N.
     The size of R is the common size of the input parameters.  A scalar
     input functions as a constant matrix of the same size as the other
     inputs.

     The parameters T, M, and N must be positive integers with M and N
     not greater than T.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: hygecdf, hygeinv, hygepdf, hygestat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 51
Random arrays from the hypergeometric distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
iwishpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 675
 -- statistics: Y = iwishpdf (W, TAU, DF, LOG_Y=false)

     Compute the probability density function of the inverse Wishart
     distribution.

     Inputs: A P x P matrix W where to find the PDF and the P x P
     positive definite scale matrix TAU and scalar degrees of freedom
     parameter DF characterizing the inverse Wishart distribution.  (For
     the density to be finite, need DF > (P - 1).)  If the flag LOG_Y is
     set, return the log probability density - this helps avoid
     underflow when the numerical value of the density is very small.

     Output: Y is the probability density of Wishart(SIGMA, DF) at W.

     See also: iwishrnd, wishpdf, wishrnd.


# name: <cell-element>
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Compute the probability density function of the inverse Wishart
distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
iwishrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1080
 -- statistics: [W, DI] = iwishrnd (PSI, DF, DI, N=1)

     Return a random matrix sampled from the inverse Wishart
     distribution with given parameters.

     Inputs: the P x P positive definite matrix TAU and scalar degrees
     of freedom parameter DF (and optionally the transposed Cholesky
     factor DI of SIGMA = 'inv(Tau)').

     DF can be non-integer as long as DF > D

     Output: a random P x P matrix W from the inverse Wishart(TAU, DF)
     distribution.  ('inv(W)' is from the Wishart('inv(Tau)', DF)
     distribution.)  If N > 1, then W is P x P x N and holds N such
     random matrices.  (Optionally, the transposed Cholesky factor DI of
     SIGMA is also returned.)

     Averaged across many samples, the mean of W should approach TAU /
     (DF - P - 1).

     References
     ----------

       1. Yu-Cheng Ku and Peter Bloomfield (2010), Generating Random
          Wishart Matrices with Fractional Degrees of Freedom in OX,
          http://www.gwu.edu/~forcpgm/YuChengKu-030510final-WishartYu-ChengKu.pdf

     See also: iwishpdf, wishpdf, wishrnd.


# name: <cell-element>
# type: sq_string
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# length: 80
Return a random matrix sampled from the inverse Wishart distribution
with giv...



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
jsucdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 580
 -- statistics: P = jsucdf (X)
 -- statistics: P = jsucdf (X, ALPHA1)
 -- statistics: P = jsucdf (X, ALPHA1, ALPHA2)

     Johnson SU cumulative distribution function (CDF).

     For each element of X, return the cumulative distribution functions
     (CDF) at X of the Johnson SU distribution with shape parameters
     ALPHA1 and ALPHA2.  The size of P is the common size of the input
     arguments X, ALPHA1, and ALPHA2.  A scalar input functions as a
     constant matrix of the same size as the other

     Default values are ALPHA1 = 1, ALPHA2 = 1.

     See also: jsupdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 50
Johnson SU cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
jsupdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 572
 -- statistics: Y = jsupdf (X)
 -- statistics: Y = jsupdf (X, ALPHA1)
 -- statistics: Y = jsupdf (X, ALPHA1, ALPHA2)

     Johnson SU probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the Johnson SU distribution with shape parameters
     ALPHA1 and ALPHA2.  The size of P is the common size of the input
     arguments X, ALPHA1, and ALPHA2.  A scalar input functions as a
     constant matrix of the same size as the other

     Default values are ALPHA1 = 1, ALPHA2 = 1.

     See also: jsucdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 46
Johnson SU probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 11
laplace_cdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 705
 -- statistics: P = laplace_cdf (X)
 -- statistics: P = laplace_cdf (X, MU)
 -- statistics: P = laplace_cdf (X, MU, BETA)

     Laplace cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Laplace distribution with a location parameter MU
     and a scale parameter (i.e.  "diversity") BETA.  The size of P is
     the common size of X, MU, and BETA.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     Default values are MU = 0, BETA = 1.  Both parameters must be reals
     and BETA > 0.  For BETA <= 0, NaN is returned.

     See also: laplace_inv, laplace_pdf, laplace_rnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Laplace cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 11
laplace_inv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 716
 -- statistics: X = laplace_inv (P)
 -- statistics: X = laplace_inv (P, MU)
 -- statistics: X = laplace_inv (P, MU, BETA)

     Inverse of the Laplace cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the Laplace distribution with a location parameter MU
     and a scale parameter (i.e.  "diversity") BETA.  The size of X is
     the common size of P, MU, and BETA.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     Default values are MU = 0, BETA = 1.  Both parameters must be reals
     and BETA > 0.  For BETA <= 0, NaN is returned.

     See also: laplace_inv, laplace_pdf, laplace_rnd.


# name: <cell-element>
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# length: 63
Inverse of the Laplace cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 11
laplace_pdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 697
 -- statistics: Y = laplace_pdf (X)
 -- statistics: Y = laplace_pdf (X, MU)
 -- statistics: Y = laplace_pdf (X, MU, BETA)

     Laplace probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the Laplace distribution with a location parameter MU
     and a scale parameter (i.e.  "diversity") BETA.  The size of Y is
     the common size of X, MU, and BETA.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     Default values are MU = 0, BETA = 1.  Both parameters must be reals
     and BETA > 0.  For BETA <= 0, NaN is returned.

     See also: laplace_cdf, laplace_inv, laplace_rnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 43
Laplace probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 11
laplace_rnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1048
 -- statistics: R = laplace_rnd (MU, BETA)
 -- statistics: R = laplace_rnd (MU, BETA, ROWS)
 -- statistics: R = laplace_rnd (MU, BETA, ROWS, COLS, ...)
 -- statistics: R = laplace_rnd (MU, BETA, [SZ])

     Random arrays from the Laplace distribution.

     'R = laplace_rnd (MU, BETA)' returns an array of random numbers
     chosen from the Laplace distribution with parameters MU and BETA.
     The size of R is the common size of MU and BETA.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.  Both parameters must be reals and BETA > 0.  For BETA <=
     0, NaN is returned.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: laplace_cdf, laplace_inv, laplace_rnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 44
Random arrays from the Laplace distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 12
logistic_cdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 678
 -- statistics: P = logistic_cdf (X)
 -- statistics: P = logistic_cdf (X, MU)
 -- statistics: P = logistic_cdf (X, MU, SCALE)

     Logistic cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the logistic distribution with mean MU and scale
     parameter SCALE.  The size of P is the common size of X, MU, and
     SCALE.  A scalar input functions as a constant matrix of the same
     size as the other inputs.

     Default values are MU = 0, BETA = 1.  Both parameters must be reals
     and BETA > 0.  For BETA <= 0, NaN is returned.

     See also: logistic_inv, logistic_pdf, logistic_rnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Logistic cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 12
logistic_inv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 690
 -- statistics: X = logistic_inv (P)
 -- statistics: X = logistic_inv (P, MU)
 -- statistics: X = logistic_inv (P, MU, SCALE)

     Inverse of the logistic cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the logistic distribution with mean MU and scale
     parameter SCALE.  The size of P is the common size of P, MU, and
     SCALE.  A scalar input functions as a constant matrix of the same
     size as the other inputs.

     Default values are MU = 0, SCALE = 1.  Both parameters must be
     reals and BETA > 0.  For BETA <= 0, NaN is returned.

     See also: logistic_cdf, logistic_pdf, logistic_rnd.


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Inverse of the logistic cumulative distribution function (iCDF).



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logistic_pdf


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 -- statistics: Y = logistic_pdf (X)
 -- statistics: Y = logistic_pdf (X, MU)
 -- statistics: Y = logistic_pdf (X, MU, SCALE)

     Logistic probability density function (PDF).

     For each element of X, compute the PDF at X of the logistic
     distribution with mean MU and scale parameter SCALE.  The size of Y
     is the common size of X, MU, and SCALE.  A scalar input functions
     as a constant matrix of the same size as the other inputs.

     Default values are MU = 0, SCALE = 1.  Both parameters must be
     reals and BETA > 0.  For BETA <= 0, NaN is returned.

     See also: logistic_cdf, logistic_inv, logistic_rnd.


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Logistic probability density function (PDF).



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logistic_rnd


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 -- statistics: R = logistic_rnd (MU, SCALE)
 -- statistics: R = logistic_rnd (MU, SCALE, ROWS)
 -- statistics: R = logistic_rnd (MU, SCALE, ROWS, COLS, ...)
 -- statistics: R = logistic_rnd (MU, SCALE, [SZ])

     Random arrays from the logistic distribution.

     'R = logistic_rnd (MU, SCALE)' returns an array of random numbers
     chosen from the logistic distribution with parameters MU and SCALE.
     The size of R is the common size of MU and SCALE.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.  Both parameters must be reals and SCALE > 0.  For SCALE <=
     0, NaN is returned.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: logistic_cdf, logistic_inv, logistic_pdf.


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Random arrays from the logistic distribution.



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logncdf


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 -- statistics: P = logncdf (X)
 -- statistics: P = logncdf (X, MU)
 -- statistics: P = logncdf (X, MU, SIGMA)
 -- statistics: P = logncdf (..., "upper")
 -- statistics: [P, PLO, PUP] = logncdf (X, MU, SIGMA, PCOV)
 -- statistics: [P, PLO, PUP] = logncdf (X, MU, SIGMA, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = logncdf (..., "upper")

     Lognormal cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the lognormal distribution with mean MU and standard
     deviation SIGMA.  The size of P is the common size of X, MU and
     SIGMA.  A scalar input functions as a constant matrix of the same
     size as the other inputs.

     Default values are MU = 0, SIGMA = 1.

     When called with three output arguments, '[P, PLO, PUP]' it
     computes the confidence bounds for P when the input parameters MU
     and SIGMA are estimates.  In such case, PCOV, a 2-by-2 matrix
     containing the covariance matrix of the estimated parameters, is
     necessary.  Optionally, ALPHA has a default value of 0.05, and
     specifies 100 * (1 - ALPHA)% confidence bounds.  PLO and PUP are
     arrays of the same size as P containing the lower and upper
     confidence bounds.

     '[...] = logncdf (..., "upper")' computes the upper tail
     probability of the lognormal distribution.

     See also: logninv, lognpdf, lognrnd, lognstat.


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Lognormal cumulative distribution function (CDF).



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logninv


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 -- statistics: X = logninv (P)
 -- statistics: X = logninv (P, MU)
 -- statistics: X = logninv (P, MU, SIGMA)

     Inverse of the lognormal cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the lognormal distribution with parameters MU and
     SIGMA.  The size of P is the common size of P, MU, and SIGMA.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     If a random variable follows this distribution, its logarithm is
     normally distributed with mean MU and standard deviation SIGMA.

     Default values are MU = 0, SIGMA = 1.  Both parameters must be
     reals and SIGMA > 0.  For SIGMA <= 0, NaN is returned.

     See also: logncdf, lognpdf, lognrnd, lognstat.


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Inverse of the lognormal cumulative distribution function (iCDF).



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lognpdf


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 -- statistics: Y = laplace_pdf (X)
 -- statistics: Y = laplace_pdf (X, MU)
 -- statistics: Y = laplace_pdf (X, MU, SIGMA)

     Lognormal probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the lognormal distribution with parameters MU and
     SIGMA.  The size of P is the common size of P, MU, and SIGMA.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     If a random variable follows this distribution, its logarithm is
     normally distributed with mean MU and standard deviation SIGMA.

     Default values are MU = 0, SIGMA = 1.  Both parameters must be
     reals and SIGMA > 0.  For SIGMA <= 0, NaN is returned.

     See also: logncdf, logninv, lognrnd, lognstat.


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Lognormal probability density function (PDF).



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lognrnd


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 -- statistics: R = lognrnd (MU, SIGMA)
 -- statistics: R = lognrnd (MU, SIGMA, ROWS)
 -- statistics: R = lognrnd (MU, SIGMA, ROWS, COLS, ...)
 -- statistics: R = lognrnd (MU, SIGMA, [SZ])

     Random arrays from the lognormal distribution.

     'R = laplace_rnd (MU, SIGMA)' returns an array of random numbers
     chosen from the lognormal distribution with parameters MU and
     SIGMA.  The size of R is the common size of MU and SIGMA.  A scalar
     input functions as a constant matrix of the same size as the other
     inputs.  Both parameters must be reals and SIGMA > 0.  For SIGMA <=
     0, NaN is returned.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: logncdf, logninv, lognpdf, lognstat.


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Random arrays from the lognormal distribution.



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mnpdf


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 -- statistics: Y = mnpdf (X, PK)

     Multinomial probability density function (PDF).

     Arguments
     ---------

        * X is vector with a single sample of a multinomial distribution
          with parameter PK or a matrix of random samples from
          multinomial distributions.  In the latter case, each row of X
          is a sample from a multinomial distribution with the
          corresponding row of PK being its parameter.

        * PK is a vector with the probabilities of the categories or a
          matrix with each row containing the probabilities of a
          multinomial sample.

     Return values
     -------------

        * Y is a vector of probabilites of the random samples X from the
          multinomial distribution with corresponding parameter PK.  The
          parameter N of the multinomial distribution is the sum of the
          elements of each row of X.  The length of Y is the number of
          columns of X.  If a row of PK does not sum to '1', then the
          corresponding element of Y will be 'NaN'.

     Examples
     --------

          x = [1, 4, 2];
          pk = [0.2, 0.5, 0.3];
          y = mnpdf (x, pk);

          x = [1, 4, 2; 1, 0, 9];
          pk = [0.2, 0.5, 0.3; 0.1, 0.1, 0.8];
          y = mnpdf (x, pk);

     References
     ----------

       1. Wendy L. Martinez and Angel R. Martinez.  'Computational
          Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
          Chapman & Hall/CRC, 2001.

       2. Merran Evans, Nicholas Hastings and Brian Peacock.
          'Statistical Distributions'.  pages 134-136, Wiley, New York,
          third edition, 2000.

     See also: mnrnd.


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Multinomial probability density function (PDF).



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mnrnd


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 -- statistics: R = mnrnd (N, PK)
 -- statistics: R = mnrnd (N, PK, S)

     Random arrays from the multinomial distribution.

     Arguments
     ---------

        * N is the first parameter of the multinomial distribution.  N
          can be scalar or a vector containing the number of trials of
          each multinomial sample.  The elements of N must be
          non-negative integers.

        * PK is the second parameter of the multinomial distribution.
          PK can be a vector with the probabilities of the categories or
          a matrix with each row containing the probabilities of a
          multinomial sample.  If PK has more than one row and N is
          non-scalar, then the number of rows of PK must match the
          number of elements of N.

        * S is the number of multinomial samples to be generated.  S
          must be a non-negative integer.  If S is specified, then N
          must be scalar and PK must be a vector.

     Return values
     -------------

        * R is a matrix of random samples from the multinomial
          distribution with corresponding parameters N and PK.  Each row
          corresponds to one multinomial sample.  The number of columns,
          therefore, corresponds to the number of columns of PK.  If S
          is not specified, then the number of rows of R is the maximum
          of the number of elements of N and the number of rows of PK.
          If a row of PK does not sum to '1', then the corresponding row
          of R will contain only 'NaN' values.

     Examples
     --------

          n = 10;
          pk = [0.2, 0.5, 0.3];
          r = mnrnd (n, pk);

          n = 10 * ones (3, 1);
          pk = [0.2, 0.5, 0.3];
          r = mnrnd (n, pk);

          n = (1:2)';
          pk = [0.2, 0.5, 0.3; 0.1, 0.1, 0.8];
          r = mnrnd (n, pk);

     References
     ----------

       1. Wendy L. Martinez and Angel R. Martinez.  'Computational
          Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
          Chapman & Hall/CRC, 2001.

       2. Merran Evans, Nicholas Hastings and Brian Peacock.
          'Statistical Distributions'.  pages 134-136, Wiley, New York,
          third edition, 2000.

     See also: mnpdf.


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Random arrays from the multinomial distribution.



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mvncdf


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 -- statistics: Y = mvncdf (X)
 -- statistics: Y = mvncdf (X, MU, SIGMA)
 -- statistics: Y = mvncdf (X_LO, X_UP, MU, SIGMA)
 -- statistics: Y = mvncdf (..., OPTIONS)
 -- statistics: [Y, ERR] = mvncdf (...)

     Multivariate normal cumulative distribution function (CDF).

     'Y = mvncdf (X)' returns cumulative probability of the multivariate
     normal distribution evaluated at each row of X with zero mean and
     an identity covariance matrix.  The row of matrix X correspond to
     observations and its columns to variables.  The return argument Y
     is a column vector with the same number of rows as in X.

     'Y = mvncdf (X, MU, SIGMA)' returns cumulative probability of the
     multivariate normal distribution evaluated at each row of X with
     mean MU and a covariance matrix SIGMA.  MU can be either a scalar
     (the same of every variable) or a row vector with the same number
     of elements as the number of variables in X.  SIGMA covariance
     matrix may be specified a row vector if it only contains variances
     along its diagonal and zero covariances of the diagonal.  In such a
     case, the diagonal vector SIGMA must have the same number of
     elements as the number of variables (columns) in X.  If you only
     want to specify sigma, you can pass an empty matrix for MU.

     The multivariate normal cumulative probability at X is defined as
     the probability that a random vector V, distributed as multivariate
     normal, will fall within the semi-infinite rectangle with upper
     limits defined by X.
        * Pr{V(1)<=X(1), V(2)<=X(2), ...  V(D)<=X(D)}.

     'Y = mvncdf (X_LO, X_HI, MU, SIGMA)' returns the multivariate
     normal cumulative probability evaluated over the rectangle
     (hyper-rectangle for multivariate data in X) with lower and upper
     limits defined by X_LO and X_HI respectively.

     '[Y, ERR] = mvncdf (...)' also returns an error estimate ERR in Y.

     'Y = mvncdf (..., OPTIONS)' specifies the structure, which controls
     specific parameters for the numerical integration for numltivariate
     cases.  The required fieds are:

     "TolFun"       -- Maximum absolute error tolerance.  Default is 1e-8
                    for D == 1 | 3, 1e-4 for D > 4.  Note that for bivariate
                    normal cdf, the Octave implementation has a presicion of
                    more than 1e-10.
     "MaxFunEvals"  -- Maximum number of integrand evaluations.  Default is
                    1e7 for D > 4.
     "Display"      -- Display options.  Choices are "off" (default),
                    "iter", which shows the probability and estimated error
                    at each repetition, and "final", which shows the final
                    probability and related error after the integrand has
                    converged successfully.

     See also: mvncdf, mvnpdf, mvnrnd, bvncdf.


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Multivariate normal cumulative distribution function (CDF).



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mvnpdf


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 -- statistics: Y = mvnpdf (X, MU, SIGMA)

     Multivariate normal probability density function (PDF).

     'Y = mvnpdf (X)' returns the probability density of the
     multivariate normal distribution with zero mean and identity
     covariance matrix, evaluated at each row of X.  Rows of the N-by-D
     matrix X correspond to observations orpoints, and columns
     correspond to variables or coordinates.  Y is an N-by-1 vector.

     'Y = mvnpdf (X, MU)' returns the density of the multivariate normal
     distribution with mean MU and identity covariance matrix, evaluated
     at each row of X.  MU is a 1-by-D vector, or an N-by-D matrix, in
     which case the density is evaluated for each row of X with the
     corresponding row of MU.  MU can also be a scalar value, which
     MVNPDF replicates to match the size of X.

     'Y = mvnpdf (X, MU, SIGMA)' returns the density of the multivariate
     normal distribution with mean MU and covariance SIGMA, evaluated at
     each row of X.  SIGMA is a D-by-D matrix, or an D-by-D-by-N array,
     in which case the density is evaluated for each row of X with the
     corresponding page of SIGMA, i.e., 'mcvpdf' computes Y(I) using
     X(I,:) and SIGMA(:,:,I).  If the covariance matrix is diagonal,
     containing variances along the diagonal and zero covariances off
     the diagonal, SIGMA may also be specified as a 1-by-D matrix or a
     1-by-D-by-N array, containing just the diagonal.  Pass in the empty
     matrix for MU to use its default value when you want to only
     specify SIGMA.

     If X is a 1-by-D vector, 'mcvpdf' replicates it to match the
     leading dimension of MU or the trailing dimension of SIGMA.

     See also: mvncdf, mvnrnd.


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Multivariate normal probability density function (PDF).



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mvnrnd


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 -- statistics: R = mvnrnd (MU, SIGMA)
 -- statistics: R = mvnrnd (MU, SIGMA, N)
 -- statistics: R = mvnrnd (MU, SIGMA, N, T)
 -- statistics: [R, T] = mvnrnd (...)

     Random vectors from the multivariate normal distribution.

     'R = mvnrnd (MU, SIGMA)' returns an N-by-D matrix R of random
     vectors chosen from the multivariate normal distribution with mean
     vector MU and covariance matrix SIGMA.  MU is an N-by-D matrix, and
     'mvnrnd' generates each N of R using the corresponding N of MU.
     SIGMA is a D-by-D symmetric positive semi-definite matrix, or a
     D-by-D-by-N array.  If SIGMA is an array, 'mvnrnd' generates each N
     of R using the corresponding page of SIGMA, i.e., 'mvnrnd' computes
     R(I,:) using MU(I,:) and SIGMA(:,:,I).  If the covariance matrix is
     diagonal, containing variances along the diagonal and zero
     covariances off the diagonal, SIGMA may also be specified as a
     1-by-D matrix or a 1-by-D-by-N array, containing just the diagonal.
     If MU is a 1-by-D vector, 'mvnrnd' replicates it to match the
     trailing dimension of SIGMA.

     'R = mvnrnd (MU, SIGMA, N)' returns a N-by-D matrix R of random
     vectors chosen from the multivariate normal distribution with
     1-by-D mean vector MU, and D-by-D covariance matrix SIGMA.

     'R = mvnrnd (MU, SIGMA, N, T)' supplies the Cholesky factor T of
     SIGMA, so that SIGMA(:,:,J) == T(:,:,J)'*T(:,:,J) if SIGMA is a 3D
     array or SIGMA == T'*T if SIGMA is a matrix.  No error checking is
     done on T.

     '[R, T] = mvnrnd (...)' returns the Cholesky factor T, so it can be
     re-used to make later calls more efficient, although there are
     greater efficiency gains when SIGMA can be specified as a diagonal
     instead.

     See also: mvncdf, mvnpdf.


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Random vectors from the multivariate normal distribution.



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mvtcdf


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 -- statistics: P = mvtcdf (X, SIGMA, DF)
 -- statistics: P = mvtcdf (A, X, SIGMA, DF)
 -- statistics: [P, ERR] = mvtcdf (...)

     Multivariate Student's t cumulative distribution function (CDF).

     Arguments
     ---------

        * X is the upper limit for integration where each row
          corresponds to an observation.

        * SIGMA is the correlation matrix.

        * DF is the degrees of freedom.

        * A is the lower limit for integration where each row
          corresponds to an observation.  A must have the same size as
          X.

     Return values
     -------------

        * P is the cumulative distribution at each row of X and A.

        * ERR is the estimated error.

     Examples
     --------

          x = [1 2];
          sigma = [1.0 0.5; 0.5 1.0];
          df = 4;
          p = mvtcdf (x, sigma, df)

          a = [-inf 0];
          p = mvtcdf (a, x, sigma, df)

     References
     ----------

       1. Alan Genz and Frank Bretz.  Numerical Computation of
          Multivariate t-Probabilities with Application to Power
          Calculation of Multiple Constrasts.  'Journal of Statistical
          Computation and Simulation', 63, pages 361-378, 1999.

     See also: mvtcdfqmc, mvtpdf, mvtrnd.


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Multivariate Student's t cumulative distribution function (CDF).



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mvtcdfqmc


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 -- statistics: P = mvtcdfqmc (A, B, RHO, DF)
 -- statisticse: P = mvtcdfqmc (..., TOLFUN)
 -- statisticse: P = mvtcdfqmc (..., TOLFUN, MAXFUNEVALS)
 -- statistics: P = mvtcdfqmc (..., TOLFUN, MAXFUNEVALS, DISPLAY)
 -- statistics: [P, ERR] = mvncdf (...)
 -- statistics: [P, ERR, FUNEVALS] = mvncdf (...)

     Quasi-Monte-Carlo computation of the multivariate Student's t cdf.

     The QMC multivariate Student's t distribution is evaluated between
     the lower limit A and upper limit B of the hyper-rectangle with a
     correlation matrix RHO and degrees of freedom DF.

     "TolFun"       -- Maximum absolute error tolerance.  Default is 1e-4.
     "MaxFunEvals"  -- Maximum number of integrand evaluations.  Default is
                    1e7 for D > 4.
     "Display"      -- Display options.  Choices are "off" (default),
                    "iter", which shows the probability and estimated error
                    at each repetition, and "final", which shows the final
                    probability and related error after the integrand has
                    converged successfully.

     '[P, ERR, FUNEVALS] = mvncdf (...)' returns the estimated
     probability, P, an estimate of the error, ERR, and the number of
     iterations until a successful convergence is met, unless the value
     in MAXFUNEVALS was reached.

     See also: mvtcdf, mvtpdf, mvtrnd.


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Quasi-Monte-Carlo computation of the multivariate Student's t cdf.



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mvtpdf


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 -- statistics: Y = mvtpdf (X, SIGMA, DF)

     Multivariate Student's t probability density function (PDF).

     Arguments
     ---------

        * X are the points at which to find the probability, where each
          row corresponds to an observation.  (N by D matrix)

        * SIGMA is the scale matrix.  (D by D symmetric positive
          definite matrix)

        * DF is the degrees of freedom.  (scalar or N vector)

     The distribution is assumed to be centered (zero mean).

     Return values
     -------------

        * Y is the probability density for each row of X.  (N by 1
          vector)

     Examples
     --------

          x = [1 2];
          sigma = [1.0 0.5; 0.5 1.0];
          df = 4;
          y = mvtpdf (x, sigma, df)

     References
     ----------

       1. Michael Roth, On the Multivariate t Distribution, Technical
          report from Automatic Control at Linkoepings universitet,
          <http://users.isy.liu.se/en/rt/roth/student.pdf>

     See also: mvtcdf, mvtcdfqmc, mvtrnd.


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Multivariate Student's t probability density function (PDF).



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mvtrnd


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 -- statistics: R = mvtrnd (SIGMA, DF)
 -- statistics: R = mvtrnd (SIGMA, DF, N)

     Random vectors from the multivariate Student's t distribution.

     Arguments
     ---------

        * SIGMA is the matrix of correlation coefficients.  If there are
          any non-unit diagonal elements then SIGMA will be normalized,
          so that the resulting covariance of the obtained samples R
          follows: 'cov (r) = df/(df-2) * sigma ./ (sqrt (diag (sigma) *
          diag (sigma)))'.  In order to obtain samples distributed
          according to a standard multivariate t-distribution, SIGMA
          must be equal to the identity matrix.  To generate
          multivariate t-distribution samples R with arbitrary
          covariance matrix SIGMA, the following scaling might be used:
          'r = mvtrnd (sigma, df, n) * diag (sqrt (diag (sigma)))'.

        * DF is the degrees of freedom for the multivariate
          t-distribution.  DF must be a vector with the same number of
          elements as samples to be generated or be scalar.

        * N is the number of rows of the matrix to be generated.  N must
          be a non-negative integer and corresponds to the number of
          samples to be generated.

     Return values
     -------------

        * R is a matrix of random samples from the multivariate
          t-distribution with N row samples.

     Examples
     --------

          sigma = [1, 0.5; 0.5, 1];
          df = 3;
          n = 10;
          r = mvtrnd (sigma, df, n);

          sigma = [1, 0.5; 0.5, 1];
          df = [2; 3];
          n = 2;
          r = mvtrnd (sigma, df, 2);

     References
     ----------

       1. Wendy L. Martinez and Angel R. Martinez.  'Computational
          Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
          Chapman & Hall/CRC, 2001.

       2. Samuel Kotz and Saralees Nadarajah.  'Multivariate t
          Distributions and Their Applications'.  Cambridge University
          Press, Cambridge, 2004.

     See also: mvtcdf, mvtcdfqmc, mvtpdf.


# name: <cell-element>
# type: sq_string
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Random vectors from the multivariate Student's t distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
nakacdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 450
 -- statistics: P = nakacdf (X, M, W)

     Nakagami cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Nakagami distribution with shape parameter M and
     scale parameter W.  The size of P is the common size of X, M, and
     W.  A scalar input functions as a constant matrix of the same size
     as the other inputs.

     See also: nakainv, nakapdf, nakarnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Nakagami cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
nakainv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 461
 -- statistics: X = nakacdf (X, M, W)

     Inverse of the Nakagami cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the Nakagami distribution with shape parameter M and
     scale parameter W.  The size of P is the common size of X, M, and
     W.  A scalar input functions as a constant matrix of the same size
     as the other inputs.

     See also: nakacdf, nakapdf, nakarnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 64
Inverse of the Nakagami cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
nakapdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 442
 -- statistics: Y = nakapdf (X, M, W)

     Nakagami probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the Nakagami distribution with shape parameter M and
     scale parameter W.  The size of P is the common size of X, M, and
     W.  A scalar input functions as a constant matrix of the same size
     as the other inputs.

     See also: nakacdf, nakainv, nakarnd.


# name: <cell-element>
# type: sq_string
# elements: 1
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Nakagami probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
nakarnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 903
 -- statistics: R = nakarnd (M, W)
 -- statistics: R = nakarnd (M, W, ROWS)
 -- statistics: R = nakarnd (M, W, ROWS, COLS, ...)
 -- statistics: R = nakarnd (M, W, [SZ])

     Random arrays from the Nakagami distribution.

     'R = nakarnd (M, W)' returns an array of random numbers chosen from
     the Nakagami distribution with parameters M and W.  The size of R
     is the common size of M and W.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: nakacdf, nakainv, nakapdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 45
Random arrays from the Nakagami distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
nbincdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 958
 -- statistics: P = nbincdf (X, N, PS)
 -- statistics: P = nbincdf (X, N, PS, "upper")

     Negative binomial cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the negative binomial distribution with parameters N
     and PS.  The size of P is the common size of X, N, and PS.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     The algorithm uses the cumulative sums of the binomial masses.

     'P = nbincdf (X, N, PS, "upper")' computes the upper tail
     probability of the lognormal distribution.

     When N is integer this is the Pascal distribution.  When N is
     extended to real numbers this is the Polya distribution.

     The number of failures in a Bernoulli experiment with success
     probability PS before the N-th success follows this distribution.

     See also: nbininv, nbinpdf, nbinrnd, nbinstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 57
Negative binomial cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
nbininv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 745
 -- statistics: X = nbininv (P, N, PS)

     Inverse of the negative binomial cumulative distribution function
     (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the negative binomial distribution with parameters N
     and PS.  The size of X is the common size of P, N, and PS.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     When N is integer this is the Pascal distribution.  When N is
     extended to real numbers this is the Polya distribution.

     The number of failures in a Bernoulli experiment with success
     probability PS before the N-th success follows this distribution.

     See also: nbininv, nbinpdf, nbinrnd, nbinstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 73
Inverse of the negative binomial cumulative distribution function
(iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
nbinpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 721
 -- statistics: Y = nbinpdf (X, N, PS)

     Negative binomial probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the negative binomial distribution with parameters N
     and PS.  The size of P is the common size of X, N, and PS.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     When N is integer this is the Pascal distribution.  When N is
     extended to real numbers this is the Polya distribution.

     The number of failures in a Bernoulli experiment with success
     probability PS before the N-th success follows this distribution.

     See also: nbininv, nbininv, nbinrnd, nbinstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 53
Negative binomial probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
nbinrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 928
 -- statistics: R = nbinrnd (N, PS)
 -- statistics: R = nbinrnd (N, PS, ROWS)
 -- statistics: R = nbinrnd (N, PS, ROWS, COLS, ...)
 -- statistics: R = nbinrnd (N, PS, [SZ])

     Random arrays from the negative binomial distribution.

     'R = nbinrnd (N, PS)' returns an array of random numbers chosen
     from the Laplace distribution with parameters N and PS.  The size
     of R is the common size of N and PS.  A scalar input functions as a
     constant matrix of the same size as the other inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: nbininv, nbininv, nbinpdf, nbinstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 54
Random arrays from the negative binomial distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
ncfcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 779
 -- statistics: P = ncfcdf (X, DF1, DF2, DELTA)
 -- statistics: P = ncfcdf (X, DF1, DF2, DELTA, UFLAG)

     Noncentral F cumulative distribution function (CDF).

     'P = ncfcdf (X, DF1, DF2, DELTA)' returns the noncentral F cdf with
     DF degrees of freedom and noncentrality parameter DELTA at the
     values of X.

     The size of P is the common size of the input arguments.  Scalar
     input arguments X, DF1, DF2, DELTA are regarded as constant
     matrices of the same size as the other inputs.

     'P = ncfcdf (X, DF1, DF2, DELTA, "upper"' returns the upper tail
     probability of the noncentral T distribution with DF1 and DF2
     degrees of freedom and noncentrality parameter DELTA at the values
     in X.

     See also: ncfinv, ncfpdf, ncfrnd, ncfstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 52
Noncentral F cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
ncfinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 630
 -- statistics: X = ncfinv (P, DF1, DF2, DELTA)

     Inverse of the non-central F cumulative distribution function
     (iCDF).

     'X = ncfinv (P, DF1, DF2, DELTA)' the inverse of the noncentral F
     distribution with DF1 numerator degrees of freedom, DF2 denumerator
     degrees of freedom, and noncentrality parameter DELTA, at the
     probabilities of P.

     The size of X is the common size of DF and DELTA.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     'ncfinv' uses Newton's method to converge to the solution.

     See also: ncfcdf, ncfpdf, ncfrnd, ncfstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 69
Inverse of the non-central F cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
ncfpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 527
 -- statistics: Y = ncfpdf (X, DF1, DF2, DELTA)

     Noncentral F probability density function (PDF).

     'Y = ncfpdf (X, DF1, DF2, DELTA)' returns the noncentral F
     distribution with DF1 and DF2 numerator and denumerator degrees of
     freedom, respectively, and noncentrality parameter DELTA, at the
     values of X.

     The size of Y is the common size of DF and DELTA.  A scalar input
     functions as a constant matrix of the same size as the K_all
     inputs.

     See also: ncfcdf, ncfinv, ncfrnd, ncfstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Noncentral F probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
ncfrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1110
 -- statistics: R = ncfrnd (DF1, DF2, DELTA)
 -- statistics: R = ncfrnd (DF1, DF2, DELTA, ROWS, COLS, ...)
 -- statistics: R = ncfrnd (DF1, DF2, DELTA, [SZ])

     Random arrays from the noncentral F distribution.

     'X = ncfrnd (P, DF1, DF2, DELTA)' returns an array of random
     numbers chosen from the noncentral F distribution with parameters
     DF1, DF2, DELTA).  The size of R is the common size of DF1, DF2,
     and DELTA.  A scalar input functions as a constant matrix of the
     same size as the other input.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     'ncfrnd' generates values using the definition of a noncentral F
     random variable, as the ratio of a noncentral chi-square and a
     (central) chi-square.

     See also: ncfcdf, ncfinv, ncfpdf, ncfstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 49
Random arrays from the noncentral F distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nctcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 735
 -- statistics: P = nctcdf (X, DF, DELTA)
 -- statistics: P = nctcdf (X, DF, DELTA, UFLAG)

     Noncentral T cumulative distribution function (CDF).

     'P = nctcdf (X, DF, DELTA)' returns the noncentral T cdf with DF
     degrees of freedom and noncentrality parameter DELTA at the values
     of X.

     The size of P is the common size of the input arguments.  Scalar
     input arguments X, DF, DELTA are regarded as constant matrices of
     the same size as the other inputs.

     'P = nctcdf (X, DF, DELTA, "upper"' returns the upper tail
     probability of the noncentral T distribution with DF degrees of
     freedom and noncentrality parameter DELTA at the values in X.

     See also: nctinv, nctpdf, nctrnd, nctstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 52
Noncentral T cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nctinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 565
 -- statistics: X = ncx2inv (P, DF, DELTA)

     Inverse of the non-central T cumulative distribution function
     (iCDF).

     'X = nctinv (P, DF, DELTA)' returns the inverse of the noncentral T
     cdf with DF degrees of freedom and noncentrality parameter DELTA,
     at the probabilities of P.

     The size of X is the common size of DF and DELTA.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     'nctinv' uses Newton's method to converge to the solution.

     See also: nctcdf, nctpdf, nctrnd, nctstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 69
Inverse of the non-central T cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nctpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 472
 -- statistics: Y = nctpdf (X, DF, DELTA)

     Noncentral Τ probability density function (pdf).

     'Y = nctpdf (X, DF, DELTA)' returns the noncentral T distribution
     with DF degrees of freedom and noncentrality parameter DELTA, at
     the values of X.

     The size of Y is the common size of DF and DELTA.  A scalar input
     functions as a constant matrix of the same size as the K_all
     inputs.

     See also: nctcdf, nctinv, nctrnd, nctstat, tpdf, pdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 49
Noncentral Τ probability density function (pdf).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nctrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1091
 -- statistics: R = nctrnd (DF, DELTA)
 -- statistics: R = nctrnd (DF, DELTA, ROWS, COLS, ...)
 -- statistics: R = nctrnd (DF, DELTA, [SZ])

     Random arrays from the noncentral T distribution.

     'X = nctrnd (P, DF, DELTA)' returns an array of random numbers
     chosen from the noncentral F distribution with parameters DF1, DF2,
     DELTA).  The size of R is the common size of DF1, DF2, and DELTA.
     A scalar input functions as a constant matrix of the same size as
     the other input.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     'nctrnd' generates values using the definition of a noncentral T
     random variable, as the ratio of a normal with non-zero mean and
     the sqrt of a chi-square.

     See also: nctcdf, nctinv, nctpdf, nctstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 49
Random arrays from the noncentral T distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
ncx2cdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 775
 -- statistics: P = ncx2cdf (X, DF, DELTA)
 -- statistics: P = ncx2cdf (X, DF, DELTA, UFLAG)

     Noncentral Chi-Square cumulative distribution function (CDF).

     'P = ncx2cdf (X, DF, DELTA)' returns the noncentral chi-square cdf
     with DF degrees of freedom and noncentrality parameter DELTA at the
     values of X.

     The size of P is the common size of the input arguments.  Scalar
     input arguments X, DF, DELTA are regarded as constant matrices of
     the same size as the other inputs.

     'P = ncx2cdf (X, DF, DELTA, "upper"' returns the upper tail
     probability of the noncentral chi-square distribution with DF
     degrees of freedom and noncentrality parameter DELTA at the values
     in X.

     See also: ncx2inv, ncx2pdf, ncx2rnd, ncx2stat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 61
Noncentral Chi-Square cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
ncx2inv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 598
 -- statistics: X = ncx2inv (P, DF, DELTA)

     Inverse of the non-central chi-square cumulative distribution
     function (iCDF).

     'X = ncx2inv (P, DF, DELTA)' returns the inverse of the noncentral
     chi-square distribution with DF degrees of freedom and
     noncentrality parameter DELTA, at the probabilities of P.

     The size of X is the common size of DF and DELTA.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     'ncx2inv' uses Newton's method to converge to the solution.

     See also: ncx2cdf, ncx2pdf, ncx2rnd, ncx2stat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 78
Inverse of the non-central chi-square cumulative distribution function
(iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
ncx2pdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 510
 -- statistics: Y = ncx2pdf (X, DF, DELTA)

     Noncentral Chi-Square probability distribution function (PDF).

     'Y = ncx2pdf (X, DF, DELTA)' returns the noncentral chi-square pdf
     with DF degrees of freedom and noncentrality parameter DELTA at the
     values of X.

     The size of Y is the common size of the input arguments.  Scalar
     input arguments X, DF, DELTA are regarded as constant matrices of
     the same size as the other inputs.

     See also: ncx2cdf, ncx2inv, ncx2rnd, ncx2stat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 62
Noncentral Chi-Square probability distribution function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
ncx2rnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 966
 -- statistics: R = ncx2rnd (DF, DELTA)
 -- statistics: R = ncx2rnd (DF, DELTA, ROWS, COLS, ...)
 -- statistics: R = ncx2rnd (DF, DELTA, [SZ])

     Random arrays from the non-central chi-square distribution.

     'R = ncx2rnd (DF, DELTA)' returns an array of random numbers chosen
     from the non-central chi-square distribution with DF degrees of
     freedom and noncentrality parameter DELTA.  The size of R is the
     common size of DF and DELTA.  A scalar input functions as a
     constant matrix of the same size as the other input.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: ncx2cdf, ncx2inv, ncx2pdf, ncx2stat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 59
Random arrays from the non-central chi-square distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
normcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1567
 -- statistics: P = normcdf (X)
 -- statistics: P = normcdf (X, MU)
 -- statistics: P = normcdf (X, MU, SIGMA)
 -- statistics: P = normcdf (..., "upper")
 -- statistics: [P, PLO, PUP] = normcdf (X, MU, SIGMA, PCOV)
 -- statistics: [P, PLO, PUP] = normcdf (X, MU, SIGMA, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = normcdf (..., "upper")

     Normal cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the normal distribution with mean MU and standard
     deviation SIGMA.  The size of P is the common size of X, MU and
     SIGMA.  A scalar input functions as a constant matrix of the same
     size as the other inputs.

     Default values are MU = 0, SIGMA = 1.

     When called with three output arguments, '[P, PLO, PUP]' it
     computes the confidence bounds for P when the input parameters MU
     and SIGMA are estimates.  In such case, PCOV, a 2-by-2 matrix
     containing the covariance matrix of the estimated parameters, is
     necessary.  Optionally, ALPHA has a default value of 0.05, and
     specifies 100 * (1 - ALPHA)% confidence bounds.  PLO and PUP are
     arrays of the same size as P containing the lower and upper
     confidence bounds.

     '[...] = normcdf (..., "upper")' computes the upper tail
     probability of the normal distribution.  This can be used to
     compute a right-tailed p-value.  To compute a two-tailed p-value,
     use '2 * normcdf (-abs (X), MU, SIGMA)'.

     See also: norminv, normpdf, normrnd, normfit, normlike, normstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 46
Normal cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
norminv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 604
 -- statistics: X = norminv (P)
 -- statistics: X = norminv (P, MU)
 -- statistics: X = norminv (P, MU, SIGMA)

     Inverse of the normal cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the normal distribution with mean MU and standard
     deviation SIGMA.  The size of P is the common size of P, MU and
     SIGMA.  A scalar input functions as a constant matrix of the same
     size as the other inputs.

     Default values are MU = 0, SIGMA = 1.

     See also: norminv, normpdf, normrnd, normfit, normlike, normstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 62
Inverse of the normal cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
normpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 577
 -- statistics: Y = normpdf (X)
 -- statistics: Y = normpdf (X, MU)
 -- statistics: Y = (X, MU, SIGMA)

     Normal probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the normal distribution with mean MU and standard
     deviation SIGMA.  The size of P is the common size of P, MU and
     SIGMA.  A scalar input functions as a constant matrix of the same
     size as the other inputs.

     Default values are MU = 0, SIGMA = 1.

     See also: norminv, norminv, normrnd, normfit, normlike, normstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 42
Normal probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
normrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1060
 -- statistics: R = normrnd (MU, SIGMA)
 -- statistics: R = normrnd (MU, SIGMA, ROWS)
 -- statistics: R = normrnd (MU, SIGMA, ROWS, COLS, ...)
 -- statistics: R = normrnd (MU, SIGMA, [SZ])

     Random arrays from the normal distribution.

     'R = normrnd (MU, SIGMA)' returns an array of random numbers chosen
     from the normal distribution with parameters MU and SIGMA.  The
     size of R is the common size of MU and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.  Both parameters must be finite real numbers and SIGMA > 0,
     otherwise NaN is returned.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: norminv, norminv, normpdf, normfit, normlike, normstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 43
Random arrays from the normal distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
poisscdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 609
 -- statistics: P = poisscdf (X, LAMBDA)
 -- statistics: P = poisscdf (X, LAMBDA, "upper")

     Poisson cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Poisson distribution with parameter LAMBDA.  The
     size of P is the common size of X and LAMBDA.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     'P = poisscdf (X, X, LAMBDA, "upper")' computes the upper tail
     probability of the lognormal distribution.

     See also: poissinv, poisspdf, poissrnd, poisstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Poisson cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
poissinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 453
 -- statistics: X = poissinv (P, LAMBDA)

     Inverse of the Poisson cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the Poisson distribution with parameter LAMBDA.  The
     size of P is the common size of X and LAMBDA.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     See also: poisscdf, poisspdf, poissrnd, poisstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 63
Inverse of the Poisson cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
poisspdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 434
 -- statistics: Y = poisspdf (X, LAMBDA)

     Poisson probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the Poisson distribution with parameter LAMBDA.  The
     size of P is the common size of X and LAMBDA.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     See also: poisscdf, poissinv, poissrnd, poisstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 43
Poisson probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
poissrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1026
 -- statistics: R = poissrnd (LAMBDA)
 -- statistics: R = poissrnd (LAMBDA, ROWS)
 -- statistics: R = poissrnd (LAMBDA, ROWS, COLS, ...)
 -- statistics: R = poissrnd (LAMBDA, [SZ])

     Random arrays from the Poisson distribution.

     'R = normrnd (LAMBDA)' returns an array of random numbers chosen
     from the Poisson distribution with parameter LAMBDA.  The size of R
     is the common size of LAMBDA.  A scalar input functions as a
     constant matrix of the same size as the other inputs.  LAMBDA must
     be a finite real number and greater or equal to 0, otherwise NaN is
     returned.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: poisscdf, poissinv, poisspdf, poisstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 44
Random arrays from the Poisson distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
raylcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1049
 -- statistics: P = raylcdf (X)
 -- statistics: P = raylcdf (X, SIGMA)
 -- statistics: P = raylcdf (..., "upper")

     Rayleigh cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the lognormal distribution with scale parameter
     SIGMA.  The size of P is the common size of X and SIGMA.  A scalar
     input functions as a constant matrix of the same size as the other
     inputs.

     Default value is SIGMA = 1.

     '[...] = logncdf (..., "upper")' computes the upper tail
     probability of the lognormal distribution.

     References
     ----------

       1. Wendy L. Martinez and Angel R. Martinez.  'Computational
          Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
          Chapman & Hall/CRC, 2001.

       2. Athanasios Papoulis.  'Probability, Random Variables, and
          Stochastic Processes'.  pages 104 and 148, McGraw-Hill, New
          York, second edition, 1984.

     See also: raylinv, raylpdf, raylrnd, raylstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Rayleigh cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
raylinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 867
 -- statistics: X = raylinv (P, SIGMA)

     Inverse of the Rayleigh cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the Rayleigh distribution with parameter SIGMA.  The
     size of P is the common size of X and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     Default value is SIGMA = 1.

     References
     ----------

       1. Wendy L. Martinez and Angel R. Martinez.  'Computational
          Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
          Chapman & Hall/CRC, 2001.

       2. Athanasios Papoulis.  'Probability, Random Variables, and
          Stochastic Processes'.  pages 104 and 148, McGraw-Hill, New
          York, second edition, 1984.

     See also: raylcdf, raylpdf, raylrnd, raylstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 64
Inverse of the Rayleigh cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
raylpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 848
 -- statistics: Y = raylpdf (X, SIGMA)

     Rayleigh probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the Rayleigh distribution with parameter SIGMA.  The
     size of P is the common size of X and SIGMA.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     Default value is SIGMA = 1.

     References
     ----------

       1. Wendy L. Martinez and Angel R. Martinez.  'Computational
          Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
          Chapman & Hall/CRC, 2001.

       2. Athanasios Papoulis.  'Probability, Random Variables, and
          Stochastic Processes'.  pages 104 and 148, McGraw-Hill, New
          York, second edition, 1984.

     See also: raylcdf, raylinv, raylrnd, raylstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 44
Rayleigh probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
raylrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1291
 -- statistics: R = raylrnd (SIGMA)
 -- statistics: R = raylrnd (SIGMA, ROWS)
 -- statistics: R = raylrnd (SIGMA, ROWS, COLS, ...)
 -- statistics: R = raylrnd (SIGMA, [SZ])

     Random arrays from the Rayleigh distribution.

     'R = raylrnd (SIGMA)' returns an array of random numbers chosen
     from the Rayleigh distribution with scale parameter SIGMA.  The
     size of R is the size of SIGMA.  SIGMA must be a finite real number
     greater than 0, otherwise NaN is returned.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     References
     ----------

       1. Wendy L. Martinez and Angel R. Martinez.  'Computational
          Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
          Chapman & Hall/CRC, 2001.

       2. Athanasios Papoulis.  'Probability, Random Variables, and
          Stochastic Processes'.  pages 104 and 148, McGraw-Hill, New
          York, second edition, 1984.

     See also: raylcdf, raylinv, raylrnd, raylstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 45
Random arrays from the Rayleigh distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 13
stdnormal_cdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 336
 -- statistics: P = stdnormal_cdf (X)

     Standard normal cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the standard normal distribution (mean = 0, standard
     deviation = 1).

     See also: normcdf, stdnormal_inv, stdnormal_pdf, stdnormal_rnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 55
Standard normal cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 13
stdnormal_inv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 352
 -- statistics: X = stdnormal_inv (P)

     Inverse of the standard normal cumulative distribution function
     (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the standard normal distribution (mean = 0, standard
     deviation = 1).

     See also: norminv, stdnormal_cdf, stdnormal_pdf, stdnormal_rnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 71
Inverse of the standard normal cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 13
stdnormal_pdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 328
 -- statistics: Y = stdnormal_pdf (X)

     Standard normal probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the standard normal distribution (mean = 0, standard
     deviation = 1).

     See also: normpdf, stdnormal_cdf, stdnormal_inv, stdnormal_rnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 51
Standard normal probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 13
stdnormal_rnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 782
 -- statistics: R = stdnormal_rnd (ROWS)
 -- statistics: R = stdnormal_rnd (ROWS, COLS, ...)
 -- statistics: R = stdnormal_rnd ([SZ])

     Random arrays from the standard normal distribution.

     'R = raylrnd (...' returns an array of random numbers chosen from
     the standard normal distribution (mean = 0, standard deviation =
     1).

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: normrnd, stdnormal_cdf, stdnormal_inv, stdnormal_pdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 52
Random arrays from the standard normal distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
tcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 629
 -- statistics: P = tcdf (X, DF)
 -- statistics: P = tcdf (X, DF, "upper")

     Student's T cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Student's T distribution with DF degrees of
     freedom.

     The size of P is the common size of X and DF.  A scalar input
     functions as a constant matrix of the same size as the other input.

     'P = tcdf (X, DF, "upper")' computes the upper tail probability of
     the Student's T distribution with DF degrees of freedom, at the
     values in X.

     See also: tinv, tpdf, trnd, tstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 51
Student's T cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
tinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 638
 -- statistics: X = tinv (P, DF)

     Inverse of the Student's T cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the Student's T distribution with DF degrees of
     freedom.

     The size of P is the common size of X and DF.  A scalar input
     functions as a constant matrix of the same size as the other input.

     This function is analogous to looking in a table for the t-value of
     a single-tailed distribution.  For very large DF (>10000), the
     inverse of the standard normal distribution is used.

     See also: tcdf, tpdf, trnd, tstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 67
Inverse of the Student's T cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
tpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 419
 -- statistics: P = tpdf (X, DF)

     Student's T probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the Student's T distribution with DF degrees of
     freedom.

     The size of Y is the common size of X and DF.  A scalar input
     functions as a constant matrix of the same size as the other input.

     See also: tcdf, tpdf, trnd, tstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Student's T probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
tricdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 467
 -- statistics: P = tricdf (X, A, B, C)

     Triangular cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the triangular distribution with parameters A, B, and
     C on the interval [A, B].  The size of P is the common size of the
     input arguments.  A scalar input functions as a constant matrix of
     the same size as the other inputs.

     See also: triinv, tripdf, trirnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 50
Triangular cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
triinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 478
 -- statistics: X = triinv (P, A, B, C)

     Inverse of the triangular cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the triangular distribution with parameters A, B, and
     C on the interval [A, B].  The size of P is the common size of the
     input arguments.  A scalar input functions as a constant matrix of
     the same size as the other inputs.

     See also: tricdf, tripdf, trirnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 66
Inverse of the triangular cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
tripdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 459
 -- statistics: Y = tripdf (X, A, B, C)

     Triangular probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the triangular distribution with parameters A, B, and
     C on the interval [A, B].  The size of Y is the common size of the
     input arguments.  A scalar input functions as a constant matrix of
     the same size as the other inputs.

     See also: tricdf, triinv, trirnd.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 46
Triangular probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
trirnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 958
 -- statistics: R = trirnd (A, B, C)
 -- statistics: R = trirnd (A, B, C, ROWS)
 -- statistics: R = trirnd (A, B, C, ROWS, COLS, ...)
 -- statistics: R = trirnd (A, B, C, [SZ])

     Random arrays from the triangular distribution.

     'R = raylrnd (SIGMA)' returns an array of random numbers chosen
     from the triangular distribution with parameters A, B, and C on the
     interval [A, B].  The size of Y is the common size of the input
     arguments.  A scalar input functions as a constant matrix of the
     same size as the other inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ..

     See also: tricdf, triinv, tripdf.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Random arrays from the triangular distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
trnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 971
 -- statistics: R = trnd (DF)
 -- statistics: R = trnd (DF, ROWS)
 -- statistics: R = trnd (DF, ROWS, COLS, ...)
 -- statistics: R = trnd (DF, [SZ])

     Random arrays from the Student's T distribution.

     Return a matrix of random samples from the t (Student) distribution
     with DF degrees of freedom.

     'R = trnd (DF)' returns an array of random numbers chosen from the
     Student's T distribution with DF degrees of freedom.  The size of R
     is the size of DF.  DF must be a finite real number greater than 0,
     otherwise NaN is returned.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: tcdf, tpdf, tpdf, tstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Random arrays from the Student's T distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unidcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 641
 -- statistics: P = unidcdf (X, DF)
 -- statistics: P = unidcdf (X, DF, "upper")

     Discrete uniform cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of a discrete uniform distribution which assumes the
     integer values 1-DF with equal probability.  The size of P is the
     common size of X and DF.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     '[...] = logncdf (..., "upper")' computes the upper tail
     probability of the lognormal distribution.

     See also: unidinv, unidpdf, unidrnd, unidstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 56
Discrete uniform cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unidinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 503
 -- statistics: X = unidinv (P, DF)

     Inverse of the discrete uniform cumulative distribution function
     (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the discrete uniform distribution which assumes the
     integer values 1-DF with equal probability.  The size of P is the
     common size of X and DF.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     See also: unidcdf, unidpdf, unidrnd, unidstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 72
Inverse of the discrete uniform cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unidpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 641
 -- statistics: Y = unidpdf (X, DF)

     Discrete uniform probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of a discrete uniform distribution which assumes the
     integer values 1-DF with equal probability.  The size of P is the
     common size of X and DF.  A scalar input functions as a constant
     matrix of the same size as the other inputs.

     Warning: The underlying implementation uses the double class and
     will only be accurate for DF < 'flintmax' (2^{53} on IEEE 754
     compatible systems).

     See also: unidcdf, unidinv, unidrnd, unidstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 52
Discrete uniform probability density function (PDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unidrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 955
 -- statistics: R = unidrnd (DF)
 -- statistics: R = unidrnd (DF, ROWS)
 -- statistics: R = unidrnd (DF, ROWS, COLS, ...)
 -- statistics: R = unidrnd (DF, [SZ])

     Random arrays from the discrete uniform distribution.

     'R = unidrnd (DF)' returns an array of random numbers chosen from
     the discrete uniform distribution with DF degrees of freedom.  The
     size of R is the size of DF.  SIGMA must be a finite integer
     greater than 0, otherwise NaN is returned.

     DF may be a scalar or a multi-dimensional array.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: unidcdf, unidinv, unidrnd, unidstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 53
Random arrays from the discrete uniform distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unifcdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 704
 -- statistics: P = unifcdf (X)
 -- statistics: P = unifcdf (X, A)
 -- statistics: P = unifcdf (X, A, B)
 -- statistics: P = unifcdf (..., "upper")

     Uniform cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the uniform distribution on the interval [A, B].  The
     size of P is the common size of the input arguments.  A scalar
     input functions as a constant matrix of the same size as the other
     inputs.

     Default values are A = 0, B = 1.

     '[...] = unifcdf (..., "upper")' computes the upper tail
     probability of the lognormal distribution.

     See also: unifinv, unifpdf, unifrnd, unifstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Uniform cumulative distribution function (CDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unifinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 561
 -- statistics: X = unifinv (P)
 -- statistics: X = unifcdf (P, A)
 -- statistics: X = unifcdf (P, A, B)

     Inverse of the uniform cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the uniform distribution on the interval [A, B].  The
     size of X is the common size of the input arguments.  A scalar
     input functions as a constant matrix of the same size as the other
     inputs.

     Default values are A = 0, B = 1.

     See also: unifcdf, unifpdf, unifrnd, unifstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 63
Inverse of the uniform cumulative distribution function (iCDF).



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
unifpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 542
 -- statistics: Y = unifpdf (X)
 -- statistics: Y = unifpdf (X, A)
 -- statistics: Y = unifpdf (X, A, B)

     Uniform probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the uniform distribution on the interval [A, B].  The
     size of Y is the common size of the input arguments.  A scalar
     input functions as a constant matrix of the same size as the other
     inputs.

     Default values are A = 0, B = 1.

     See also: unifcdf, unifinv, unifrnd, unifstat.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 43
Uniform probability density function (PDF).



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unifrnd


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 -- statistics: R = unifrnd (A, B)
 -- statistics: R = unifrnd (A, B, ROWS)
 -- statistics: R = unifrnd (A, B, ROWS, COLS, ...)
 -- statistics: R = unifrnd (A, B, [SZ])

     Random arrays from the uniform distribution.

     'R = unifrnd (A, B)' returns an array of random numbers chosen from
     a uniform distribution on the interval [A, B].  The size of Y is
     the common size of the input arguments.  A scalar input functions
     as a constant matrix of the same size as the other inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: unifcdf, unifinv, unifpdf, unifstat.


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Random arrays from the uniform distribution.



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vmcdf


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 -- statistics: X = vmcdf (THETA)
 -- statistics: X = vmcdf (THETA, MU)
 -- statistics: X = vmcdf (THETA, MU, K)

     Von Mises probability density function (PDF).

     For each element of THETA, compute the probability density function
     (PDF) at THETA of the von Mises distribution with mean direction
     parameter MU and concentration parameter K on the interval [-pi,
     pi].  The size of X is the common size of the input arguments.  A
     scalar input functions as a constant matrix of the same same size
     as the other inputs.

     Default values are MU = 0, K = 1.  The function returns NaN for
     negative K.

     See also: vmpdf, vmrnd.


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Von Mises probability density function (PDF).



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vmpdf


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 -- statistics: Y = vmpdf (THETA)
 -- statistics: Y = vmpdf (THETA, MU)
 -- statistics: Y = vmpdf (THETA, MU, K)

     Von Mises probability density function (PDF).

     For each element of THETA, compute the probability density function
     (PDF) at THETA of the von Mises distribution with mean direction
     parameter MU and concentration parameter K on the interval [-pi,
     pi].  The size of Y is the common size of the input arguments.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     Default values are MU = 0, K = 1.  The function returns NaN for
     negative K.

     See also: vmcdf, vmrnd.


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Von Mises probability density function (PDF).



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vmrnd


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 -- statistics: R = vmrnd (MU, K)
 -- statistics: R = vmrnd (MU, K, ROWS)
 -- statistics: R = vmrnd (MU, K, ROWS, COLS, ...)
 -- statistics: R = vmrnd (MU, K, [SZ])

     Random arrays from the von Mises distribution.

     'R = vmrnd (MU, K)' returns an array of random angles chosen from a
     von Mises distribution with mean direction parameter MU and
     concentration parameter K on the interval [-pi, pi].  The size of R
     is the common size of the input arguments.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: vmcdf, vmpdf.


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Random arrays from the von Mises distribution.



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wblcdf


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 -- statistics: P = wblcdf (X)
 -- statistics: P = wblcdf (X, LAMBDA)
 -- statistics: P = wblcdf (X, LAMBDA, K)
 -- statistics: P = wblcdf (..., "upper")
 -- statistics: [P, PLO, PUP] = wblcdf (X, LAMBDA, K, PCOV)
 -- statistics: [P, PLO, PUP] = wblcdf (X, LAMBDA, K, PCOV, ALPHA)
 -- statistics: [P, PLO, PUP] = wblcdf (..., "upper")

     Weibull cumulative distribution function (CDF).

     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Weibull distribution with scale parameter LAMBDA
     and shape parameter K.  The size of P is the common size of X,
     LAMBDA and K.  A scalar input functions as a constant matrix of the
     same size as the other inputs.

     Default values are LAMBDA = 0, K = 1.

     When called with three output arguments, '[P, PLO, PUP]' it
     computes the confidence bounds for P when the input parameters
     LAMBDA and K are estimates.  In such case, PCOV, a 2-by-2 matrix
     containing the covariance matrix of the estimated parameters, is
     necessary.  Optionally, ALPHA has a default value of 0.05, and
     specifies 100 * (1 - ALPHA)% confidence bounds.  PLO and PUP are
     arrays of the same size as P containing the lower and upper
     confidence bounds.

     '[...] = wblcdf (..., "upper")' computes the upper tail probability
     of the lognormal distribution.

     See also: wblinv, wblpdf, wblrnd, wblstat, wblplot.


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Weibull cumulative distribution function (CDF).



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wblinv


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 -- statistics: X = wblinv (P)
 -- statistics: X = wblinv (P, LAMBDA)
 -- statistics: X = wblinv (P, LAMBDA, K)

     Inverse of the Weibull cumulative distribution function (iCDF).

     For each element of P, compute the quantile (the inverse of the
     CDF) at P of the Weibull distribution with parameters LAMBDA and K.
     The size of X is the common size of P, LAMBDA, and K.  A scalar
     input functions as a constant matrix of the same size as the other
     inputs.

     Default values are LAMBDA = 1, K = 1.

     See also: wblcdf, wblpdf, wblrnd, wblstat, wblplot.


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Inverse of the Weibull cumulative distribution function (iCDF).



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wblpdf


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 -- statistics: Y = wblinv (X)
 -- statistics: Y = wblinv (X, LAMBDA)
 -- statistics: Y = wblinv (X, LAMBDA, XK)

     Weibull probability density function (PDF).

     For each element of X, compute the probability density function
     (PDF) at X of the Weibull distribution with parameters LAMBDA and
     XK.  The size of Y is the common size of X, LAMBDA, and XK.  A
     scalar input functions as a constant matrix of the same size as the
     other inputs.

     Default values are LAMBDA = 1, XK = 1.

     See also: wblcdf, wblinv, wblrnd, wblstat, wblplot.


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Weibull probability density function (PDF).



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wblrnd


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 -- statistics: R = wblrnd (LAMBDA, K)
 -- statistics: R = wblrnd (LAMBDA, K, ROWS)
 -- statistics: R = wblrnd (LAMBDA, K, ROWS, COLS, ...)
 -- statistics: R = wblrnd (LAMBDA, K, [SZ])

     Random arrays from the Weibull distribution.

     'R = wblrnd (LAMBDA, K)' returns an array of random numbers chosen
     from the Weibull distribution with parameters LAMBDA and K.  The
     size of R is the common size of LAMBDA and K.  A scalar input
     functions as a constant matrix of the same size as the other
     inputs.  Both parameters must be positive reals.

     When called with a single size argument, return a square matrix
     with the dimension specified.  When called with more than one
     scalar argument the first two arguments are taken as the number of
     rows and columns and any further arguments specify additional
     matrix dimensions.  The size may also be specified with a vector of
     dimensions SZ.

     See also: wblcdf, wblinv, wblpdf, wblstat, wblplot.


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Random arrays from the Weibull distribution.



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wienrnd


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 -- statistics: R = wienrnd (T, D, N)

     Return a simulated realization of the D-dimensional Wiener Process
     on the interval [0, T].

     If D is omitted, D = 1 is used.  The first column of the return
     matrix contains time, the remaining columns contain the Wiener
     process.

     The optional parameter N defines the number of summands used for
     simulating the process over an interval of length 1.  If N is
     omitted, N = 1000 is used.


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Return a simulated realization of the D-dimensional Wiener Process on
the int...



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wishpdf


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 -- statistics: Y = wishpdf (W, SIGMA, DF, LOG_Y=false)

     Compute the probability density function of the Wishart
     distribution

     Inputs: A P x P matrix W where to find the PDF. The P x P positive
     definite matrix SIGMA and scalar degrees of freedom parameter DF
     characterizing the Wishart distribution.  (For the density to be
     finite, need DF > (P - 1).)

     If the flag LOG_Y is set, return the log probability density - this
     helps avoid underflow when the numerical value of the density is
     very small

     Output: Y is the probability density of Wishart(SIGMA, DF) at W.

     See also: wishrnd, iwishpdf, iwishrnd.


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Compute the probability density function of the Wishart distribution



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wishrnd


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 -- statistics: [W, D] = wishrnd (SIGMA, DF, D, N=1)

     Return a random matrix sampled from the Wishart distribution with
     given parameters

     Inputs: the P x P positive definite matrix SIGMA (or the
     lower-triangular Cholesky factor D of SIGMA) and scalar degrees of
     freedom parameter DF.

     DF can be non-integer as long as DF > P

     Output: a random P x P matrix W from the Wishart(SIGMA, DF)
     distribution.  If N > 1, then W is P x P x N and holds N such
     random matrices.  (Optionally, the lower-triangular Cholesky factor
     D of SIGMA is also returned.)

     Averaged across many samples, the mean of W should approach
     DF*SIGMA, and the variance of each element W_ij should approach
     DF*(SIGMA_ij^2 + SIGMA_ii*SIGMA_jj)

     References
     ----------

       1. Yu-Cheng Ku and Peter Bloomfield (2010), Generating Random
          Wishart Matrices with Fractional Degrees of Freedom in OX,
          http://www.gwu.edu/~forcpgm/YuChengKu-030510final-WishartYu-ChengKu.pdf

     See also: wishpdf, iwishpdf, iwishrnd.


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Return a random matrix sampled from the Wishart distribution with given
param...





