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### 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).

DistributionPDFCDFQuantile
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 DistributionNot 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`
Univariate Normal Distribution`normpdf``normcdf``norminv`
Pascal Distribution`nbinpdf``nbincdf``nbininv`
Poisson Distribution`poisspdf``poisscdf``poissinv`
Standard Normal Distribution`stdnormal_pdf``stdnormal_cdf``stdnormal_inv`
t (Student) Distribution`tpdf``tcdf``tinv`
Univariate Discrete Distribution`unidpdf``unidcdf``unidinv`
Uniform Distribution`unifpdf``unifcdf``unifinv`
Weibull Distribution`wblpdf``wblcdf``wblinv`
: 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.

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