— Function File: betapdf (x, a, b)

For each element of x, compute the probability density function (PDF) at x of the Beta distribution with parameters a and b.

— Function File: betacdf (x, a, b)

For each element of x, compute the cumulative distribution function (CDF) at x of the Beta distribution with parameters a and b.

— Function File: betainv (x, a, b)

For each element of x, compute the quantile (the inverse of the CDF) at x of the Beta distribution with parameters a and b.

— Function File: binopdf (x, n, p)

For each element of x, compute the probability density function (PDF) at x of the binomial distribution with parameters n and p, where n is the number of trials and p is the probability of success.

— Function File: binocdf (x, n, p)

For each element of x, compute the cumulative distribution function (CDF) at x of the binomial distribution with parameters n and p, where n is the number of trials and p is the probability of success.

— Function File: binoinv (x, n, p)

For each element of x, compute the quantile (the inverse of the CDF) at x of the binomial distribution with parameters n and p, where n is the number of trials and p is the probability of success.

— Function File: cauchy_pdf (x)
— Function File: cauchy_pdf (x, location, scale)

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 > 0. Default values are location = 0, scale = 1.

— Function File: cauchy_cdf (x)
— Function File: cauchy_cdf (x, location, scale)

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. Default values are location = 0, scale = 1.

— Function File: cauchy_inv (x)
— Function File: cauchy_inv (x, location, scale)

For each element of x, compute the quantile (the inverse of the CDF) at x of the Cauchy distribution with location parameter location and scale parameter scale. Default values are location = 0, scale = 1.

— Function File: chi2pdf (x, n)

For each element of x, compute the probability density function (PDF) at x of the chi-square distribution with n degrees of freedom.

— Function File: chi2cdf (x, n)

For each element of x, compute the cumulative distribution function (CDF) at x of the chi-square distribution with n degrees of freedom.

— Function File: chi2inv (x, n)

For each element of x, compute the quantile (the inverse of the CDF) at x of the chi-square distribution with n degrees of freedom.

— Function File: discrete_pdf (x, v, p)

For each element of x, compute the probability density function (PDF) at x of a univariate discrete distribution which assumes the values in v with probabilities p.

— Function File: discrete_cdf (x, v, p)

For each element of x, compute the cumulative distribution function (CDF) at x of a univariate discrete distribution which assumes the values in v with probabilities p.

— Function File: discrete_inv (x, v, p)

For each element of x, compute the quantile (the inverse of the CDF) at x of the univariate distribution which assumes the values in v with probabilities p.

— Function File: empirical_pdf (x, data)

For each element of x, compute the probability density function (PDF) at x of the empirical distribution obtained from the univariate sample data.

— Function File: empirical_cdf (x, data)

For each element of x, compute the cumulative distribution function (CDF) at x of the empirical distribution obtained from the univariate sample data.

— Function File: empirical_inv (x, data)

For each element of x, compute the quantile (the inverse of the CDF) at x of the empirical distribution obtained from the univariate sample data.

— Function File: exppdf (x, lambda)

For each element of x, compute the probability density function (PDF) at x of the exponential distribution with mean lambda.

— Function File: expcdf (x, lambda)

For each element of x, compute the cumulative distribution function (CDF) at x of the exponential distribution with mean lambda.

The arguments can be of common size or scalars.

— Function File: expinv (x, lambda)

For each element of x, compute the quantile (the inverse of the CDF) at x of the exponential distribution with mean lambda.

— Function File: fpdf (x, m, n)

For each element of x, compute the probability density function (PDF) at x of the F distribution with m and n degrees of freedom.

— Function File: fcdf (x, m, n)

For each element of x, compute the cumulative distribution function (CDF) at x of the F distribution with m and n degrees of freedom.

— Function File: finv (x, m, n)

For each element of x, compute the quantile (the inverse of the CDF) at x of the F distribution with m and n degrees of freedom.

— Function File: gampdf (x, a, b)

For each element of x, return the probability density function (PDF) at x of the Gamma distribution with shape parameter a and scale b.

— Function File: gamcdf (x, a, b)

For each element of x, compute the cumulative distribution function (CDF) at x of the Gamma distribution with shape parameter a and scale b.

— Function File: gaminv (x, a, b)

For each element of x, compute the quantile (the inverse of the CDF) at x of the Gamma distribution with shape parameter a and scale b.

— Function File: geopdf (x, p)

For each element of x, compute the probability density function (PDF) at x of the geometric distribution with parameter p.

— Function File: geocdf (x, p)

For each element of x, compute the cumulative distribution function (CDF) at x of the geometric distribution with parameter p.

— Function File: geoinv (x, p)

For each element of x, compute the quantile (the inverse of the CDF) at x of the geometric distribution with parameter p.

— Function File: hygepdf (x, t, m, n)

Compute the probability density function (PDF) at x of the hypergeometric distribution with parameters t, m, and n. 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.

— Function File: hygecdf (x, t, m, n)

Compute the cumulative distribution function (CDF) at x of the hypergeometric distribution with parameters t, m, and n. 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.

— Function File: hygeinv (x, t, m, n)

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. 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.

— Function File: kolmogorov_smirnov_cdf (x, tol)

Return the cumulative distribution function (CDF) at x of the Kolmogorov-Smirnov distribution,

                   Inf
          Q(x) =   SUM    (-1)^k exp (-2 k^2 x^2)
                 k = -Inf

for x > 0.

The optional parameter tol specifies the precision up to which the series should be evaluated; the default is tol = eps.

— Function File: laplace_pdf (x)

For each element of x, compute the probability density function (PDF) at x of the Laplace distribution.

— Function File: laplace_cdf (x)

For each element of x, compute the cumulative distribution function (CDF) at x of the Laplace distribution.

— Function File: laplace_inv (x)

For each element of x, compute the quantile (the inverse of the CDF) at x of the Laplace distribution.

— Function File: logistic_pdf (x)

For each element of x, compute the PDF at x of the logistic distribution.

— Function File: logistic_cdf (x)

For each element of x, compute the cumulative distribution function (CDF) at x of the logistic distribution.

— Function File: logistic_inv (x)

For each element of x, compute the quantile (the inverse of the CDF) at x of the logistic distribution.

— Function File: lognpdf (x)
— Function File: lognpdf (x, mu, sigma)

For each element of x, compute the probability density function (PDF) at x of the lognormal distribution with parameters mu and sigma. If a random variable follows this distribution, its logarithm is normally distributed with mean mu and standard deviation sigma.

Default values are mu = 1, sigma = 1.

— Function File: logncdf (x)
— Function File: logncdf (x, mu, sigma)

For each element of x, compute the cumulative distribution function (CDF) at x of the lognormal distribution with parameters mu and sigma. If a random variable follows this distribution, its logarithm is normally distributed with mean mu and standard deviation sigma.

Default values are mu = 1, sigma = 1.

— Function File: logninv (x)
— Function File: logninv (x, mu, sigma)

For each element of x, compute the quantile (the inverse of the CDF) at x of the lognormal distribution with parameters mu and sigma. If a random variable follows this distribution, its logarithm is normally distributed with mean log (mu) and variance sigma.

Default values are mu = 1, sigma = 1.

— Function File: nbinpdf (x, n, p)

For each element of x, compute the probability density function (PDF) at x of the negative binomial distribution with parameters n and p.

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 p before the n-th success follows this distribution.

— Function File: nbincdf (x, n, p)

For each element of x, compute the cumulative distribution function (CDF) at x of the negative binomial distribution with parameters n and p.

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 p before the n-th success follows this distribution.

— Function File: nbininv (x, n, p)

For each element of x, compute the quantile (the inverse of the CDF) at x of the negative binomial distribution with parameters n and p.

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 p before the n-th success follows this distribution.

— Function File: normpdf (x)
— Function File: normpdf (x, mu, sigma)

For each element of x, compute the probability density function (PDF) at x of the normal distribution with mean mu and standard deviation sigma.

Default values are mu = 0, sigma = 1.

— Function File: normcdf (x)
— Function File: normcdf (x, mu, sigma)

For each element of x, compute the cumulative distribution function (CDF) at x of the normal distribution with mean mu and standard deviation sigma.

Default values are mu = 0, sigma = 1.

— Function File: norminv (x)
— Function File: norminv (x, mu, sigma)

For each element of x, compute the quantile (the inverse of the CDF) at x of the normal distribution with mean mu and standard deviation sigma.

Default values are mu = 0, sigma = 1.

— Function File: poisspdf (x, lambda)

For each element of x, compute the probability density function (PDF) at x of the Poisson distribution with parameter lambda.

— Function File: poisscdf (x, lambda)

For each element of x, compute the cumulative distribution function (CDF) at x of the Poisson distribution with parameter lambda.

— Function File: poissinv (x, lambda)

For each element of x, compute the quantile (the inverse of the CDF) at x of the Poisson distribution with parameter lambda.

— Function File: stdnormal_pdf (x)

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

— Function File: stdnormal_cdf (x)

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

— Function File: stdnormal_inv (x)

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

— Function File: tpdf (x, n)

For each element of x, compute the probability density function (PDF) at x of the t (Student) distribution with n degrees of freedom.

— Function File: tcdf (x, n)

For each element of x, compute the cumulative distribution function (CDF) at x of the t (Student) distribution with n degrees of freedom, i.e., PROB (t(n) ≤ x).

— Function File: tinv (x, n)

For each element of x, compute the quantile (the inverse of the CDF) at x of the t (Student) distribution with n degrees of freedom. This function is analogous to looking in a table for the t-value of a single-tailed distribution.

— Function File: unidpdf (x, n)

For each element of x, compute the probability density function (PDF) at x of a discrete uniform distribution which assumes the integer values 1–n with equal probability.

Warning: The underlying implementation uses the double class and will only be accurate for n ≤ bitmax (2^53 - 1 on IEEE-754 compatible systems).

— Function File: unidcdf (x, n)

For each element of x, compute the cumulative distribution function (CDF) at x of a discrete uniform distribution which assumes the integer values 1–n with equal probability.

— Function File: unidinv (x, n)

For each element of x, compute the quantile (the inverse of the CDF) at x of the discrete uniform distribution which assumes the integer values 1–n with equal probability.

— Function File: unifpdf (x)
— Function File: unifpdf (x, a, b)

For each element of x, compute the probability density function (PDF) at x of the uniform distribution on the interval [a, b].

Default values are a = 0, b = 1.

— Function File: unifcdf (x)
— Function File: unifcdf (x, a, b)

For each element of x, compute the cumulative distribution function (CDF) at x of the uniform distribution on the interval [a, b].

Default values are a = 0, b = 1.

— Function File: unifinv (x)
— Function File: unifinv (x, a, b)

For each element of x, compute the quantile (the inverse of the CDF) at x of the uniform distribution on the interval [a, b].

Default values are a = 0, b = 1.

— Function File: wblpdf (x)
— Function File: wblpdf (x, scale)
— Function File: wblpdf (x, scale, shape)

Compute the probability density function (PDF) at x of the Weibull distribution with scale parameter scale and shape parameter shape which is given by

             shape * scale^(-shape) * x^(shape-1) * exp (-(x/scale)^shape)

for x ≥ 0.

Default values are scale = 1, shape = 1.

— Function File: wblcdf (x)
— Function File: wblcdf (x, scale)
— Function File: wblcdf (x, scale, shape)

Compute the cumulative distribution function (CDF) at x of the Weibull distribution with scale parameter scale and shape parameter shape, which is

          1 - exp (-(x/scale)^shape)

for x ≥ 0.

Default values are scale = 1, shape = 1.

— Function File: wblinv (x)
— Function File: wblinv (x, scale)
— Function File: wblinv (x, scale, shape)

Compute the quantile (the inverse of the CDF) at x of the Weibull distribution with scale parameter scale and shape parameter shape.

Default values are scale = 1, shape = 1.