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### 25.6 Distributions

Octave has functions for computing the Probability Density Function (PDF), the Cumulative Distribution function (CDF), and the quantile (the inverse of the CDF) of a large number of distributions.

The following table summarizes the supported distributions (in alphabetical order).

Distribution | PDF | CDF | Quantile
Beta Distribution | `betapdf` | `betacdf` | `betainv`
Binomial Distribution | `binopdf` | `binocdf` | `binoinv`
Cauchy Distribution | `cauchy_pdf` | `cauchy_cdf` | `cauchy_inv`
Chi-Square Distribution | `chi2pdf` | `chi2cdf` | `chi2inv`
Univariate Discrete Distribution | `discrete_pdf` | `discrete_cdf` | `discrete_inv`
Empirical Distribution | `empirical_pdf` | `empirical_cdf` | `empirical_inv`
Exponential Distribution | `exppdf` | `expcdf` | `expinv`
F Distribution | `fpdf` | `fcdf` | `finv`
Gamma Distribution | `gampdf` | `gamcdf` | `gaminv`
Geometric Distribution | `geopdf` | `geocdf` | `geoinv`
Hypergeometric Distribution | `hygepdf` | `hygecdf` | `hygeinv`
Kolmogorov Smirnov Distribution | Not Available | `kolmogorov_smirnov_cdf` | Not Available
Laplace Distribution | `laplace_pdf` | `laplace_cdf` | `laplace_inv`
Logistic Distribution | `logistic_pdf` | `logistic_cdf` | `logistic_inv`
Log-Normal Distribution | `lognpdf` | `logncdf` | `logninv`
Pascal Distribution | `nbinpdf` | `nbincdf` | `nbininv`
Univariate Normal Distribution | `normpdf` | `normcdf` | `norminv`
Poisson Distribution | `poisspdf` | `poisscdf` | `poissinv`
t (Student) Distribution | `tpdf` | `tcdf` | `tinv`
Univariate Discrete Distribution | `unidpdf` | `unidcdf` | `unidinv`
Uniform Distribution | `unifpdf` | `unifcdf` | `unifinv`
Weibull Distribution | `wblpdf` | `wblcdf` | `wblinv`

— Function File: betacdf (x, a, b)

> For each element of x, returns the CDF at x of the beta distribution with parameters a and b, i.e., PROB (beta (a, b) <= x).

— Function File: betainv (x, a, b)

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

— Function File: betapdf (x, a, b)

> For each element of x, returns the PDF at x of the beta distribution with parameters a and b.

— Function File: binocdf (x, n, p)

> For each element of x, compute the CDF at x of the binomial distribution with parameters n and p.

— Function File: binoinv (x, n, p)

> For each element of x, compute the quantile at x of the binomial distribution with parameters n and p.

— 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.

— Function File: cauchy_cdf (x, lambda, sigma)

> For each element of x, compute the cumulative distribution function (CDF) at x of the Cauchy distribution with location parameter lambda and scale parameter sigma. Default values are lambda = 0, sigma = 1.

— Function File: cauchy_inv (x, lambda, sigma)

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

— Function File: cauchy_pdf (x, lambda, sigma)

> For each element of x, compute the probability density function (PDF) at x of the Cauchy distribution with location parameter lambda and scale parameter sigma > 0. Default values are lambda = 0, sigma = 1.

— Function File: chi2cdf (x, n)

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

— Function File: chisquare_pdf (x, n)

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

— 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 component 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: 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: 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: 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: 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 scalar.

— 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: exppdf (x, lambda)

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

— Function File: fcdf (x, m, n)

> For each element of x, compute the CDF at x of the F distribution with m and n degrees of freedom, i.e., PROB (F (m, n) <= x).

— Function File: finv (x, m, n)

> For each component of x, compute the quantile (the inverse of the CDF) at x of the F distribution with parameters m and n.

— 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: gamcdf (x, a, b)

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

> See also: gamma, gammaln, gammainc, gampdf, gaminv, gamrnd.

— Function File: gaminv (x, a, b)

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

> See also: gamma, gammaln, gammainc, gampdf, gamcdf, gamrnd.

— Function File: gampdf (x, a, b)

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

> See also: gamma, gammaln, gammainc, gamcdf, gaminv, gamrnd.

— Function File: geocdf (x, p)

> For each element of x, compute the CDF at x of the geometric distribution with parameter p.

— Function File: geoinv (x, p)

> For each element of x, compute the quantile at x of the geometric distribution with parameter p.

— 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: 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 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 at x of the hypergeometric distribution with parameters t, m, and n.

> The parameters t, m, and n must positive integers with m and n not greater than t.

— 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 arguments must be of common size or scalar.

— Function File: kolmogorov_smirnov_cdf (x, tol)

> Return the 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_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: laplace_pdf (x)

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

— Function File: logistic_cdf (x)

> For each component of x, compute the CDF at x of the logistic distribution.

— Function File: logistic_inv (x)

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

— Function File: logistic_pdf (x)

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

— 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, 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: 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: nbincdf (x, n, p)

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

> 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 at x of the Pascal (negative binomial) distribution with parameters n and p.

> The number of failures in a Bernoulli experiment with success probability p before the n-th success follows this distribution.

— Function File: nbinpdf (x, n, p)

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

> The number of failures in a Bernoulli experiment with success probability p before the n-th success follows this distribution.

— Function File: normcdf (x, m, s)

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

> Default values are m = 0, s = 1.

— Function File: norminv (x, m, s)

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

> Default values are m = 0, s = 1.

— Function File: normpdf (x, m, s)

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

> Default values are m = 0, s = 1.

— 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 component of x, compute the quantile (the inverse of the CDF) at x of the Poisson distribution with parameter lambda.

— 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: 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 probability value x, compute the inverse of the cumulative distribution function (CDF) of the t (Student) distribution with degrees of freedom n. This function is analogous to looking in a table for the t-value of a single-tailed distribution.

— 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: unidcdf (x, v)

> 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 equal probability.

— Function File: unidinv (x, v)

> For each component of x, compute the quantile (the inverse of the CDF) at x of the univariate discrete distribution which assumes the values in v with equal probability

— Function File: unidpdf (x, v)

> 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 equal probability.

— Function File: unifcdf (x, a, b)

> Return the CDF at x of the uniform distribution on [a, b], i.e., PROB (uniform (a, b) <= x).

> Default values are a = 0, b = 1.

— 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 [a, b].

> Default values are a = 0, b = 1.

— Function File: unifpdf (x, a, b)

> For each element of x, compute the PDF at x of the uniform distribution on [a, b].

> Default values are a = 0, b = 1.

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

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

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

> for x >= 0.

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

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

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

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

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

> for x > 0.