26.5 Distributions

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

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

: 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.

: 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.

: 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.

: 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.

: 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.

: 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.

: cauchy_pdf (x)

: 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.

: cauchy_cdf (x)

: 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.

: cauchy_inv (x)

: 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.

: 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.

: 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.

: 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.

: 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.

: 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.

: 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.

: 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.

: 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.

: 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.

: exppdf (x, lambda)

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

: 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.

: 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.

: 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.

: 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.

: 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.

: 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.

: 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.

: 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.

: geopdf (x, p)

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

The geometric distribution models the number of failures (x-1) of a Bernoulli trial with probability p before the first success (x).

: geocdf (x, p)

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

The geometric distribution models the number of failures (x-1) of a Bernoulli trial with probability p before the first success (x).

: 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.

The geometric distribution models the number of failures (x-1) of a Bernoulli trial with probability p before the first success (x).

: 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.

: 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.

: 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.

: kolmogorov_smirnov_cdf (x, tol)

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

This is defined as

         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.

: laplace_pdf (x)

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

: laplace_cdf (x)

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

: laplace_inv (x)

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

: logistic_pdf (x)

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

: logistic_cdf (x)

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

: logistic_inv (x)

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

: lognpdf (x)

: 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 = 0, sigma = 1.

: logncdf (x)

: 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 = 0, sigma = 1.

: logninv (x)

: 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 mu and standard deviation sigma.

Default values are mu = 0, sigma = 1.

: 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.

: 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.

: 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.

: normpdf (x)

: 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.

: normcdf (x)

: 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.

: norminv (x)

: 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.

: poisspdf (x, lambda)

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

: poisscdf (x, lambda)

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

: 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.

: 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).

: 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).

: 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).

: 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.

: 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.

: 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.

: 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 < flintmax (2^{53} on IEEE 754 compatible systems).

: 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.

: 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.

: unifpdf (x)

: 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.

: unifcdf (x)

: 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.

: unifinv (x)

: 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.

: wblpdf (x)

: wblpdf (x, scale)

: wblpdf (x, scale, shape)

Compute the probability density function (PDF) at x of the Weibull distribution with scale parameter scale and shape parameter shape.

This is given by

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

for x ≥ 0.

Default values are scale = 1, shape = 1.

: wblcdf (x)

: wblcdf (x, scale)

: wblcdf (x, scale, shape)

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

This is defined as

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

for x ≥ 0.

Default values are scale = 1, shape = 1.

: wblinv (x)

: wblinv (x, scale)

: 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.