#### Distributions

##### Introduction

Commonly encountered distributions are built into BUGS, as described on this page. Further distributions are included in GeoBUGS and ReliaBUGS: see the

Spatial distributions,

Temporal distributions and

Reliability distributions pages for details of these.

If a distribution is not built into BUGS, distributions specified by a log-likelihood can also be used: see

Generic sampling distributions for details.

##### Binomial

The Binomial distribution is defined by the pmf

In the BUGS language it is used as

`r ~ dbin(p, n)`

##### Binomial

The Binomial distribution is defined by the pmf

In the BUGS language it is used as

`r ~ dbin(p, n)`

##### Categorical

The Categorical distribution is defined by the pmf

In the BUGS language it is used as

`r ~ dcat(p[])`

##### Negative Binomial

The Negative Binomial distribution is defined by the pmf

In the BUGS language it is used as

`x ~ dnegbin(p, r)`

##### Poisson

The Poisson distribution is defined by the pmf

In the BUGS language it is used as

`r ~ dpois(lambda)`

##### Non-central hypergeometric

The Non-central hypergeometic distribution is defined by the pmf

In the BUGS language it is used as

`x ~ dhyper(n, m, N, psi)`

##### Beta

The Beta distribution is defined by the pdf

In the BUGS language it is used as

`p ~ dbeta(a, b)`

##### Chi-squared

The Chi-squared distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dchisqr(k)`

##### Double Exponential

The Double Exponential distribution is defined by the pdf

In the BUGS language it is used as

`x ~ ddexp(mu, tau)`

##### Exponential

The Exponential distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dexp(lambda)`

##### Flat

The improper Flat distribution has a constant value for all x. It is not a proper distribution.

In the BUGS language it is used as

`x ~ dflat()`

##### Gamma

The Gamma distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dgamma(r, mu)`

##### Generalized extreme value

The Generlized extreme value distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dgev(mu, sigma, eta)`

##### Generalized F

The Generalized F distribution is defined by the pdf

It reduces to the standard F for mu=0, tau=1. In the BUGS language it is used as

`x ~ df(n, m, mu, tau)`

##### Generalized Gamma

The Generalized Gamma distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dggamma(r, mu, beta)`

##### Generalized Pareto

The Generalized Pareto distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dgpar(mu, sigma, eta)`

##### Generic log-likelihood distribution

The generic log-likelihood distribution is defined by the pdf exp(lambda). It allows generic log-likelihoods to be used in BUGS. See

Generic sampling distributions for details. Note it does not depend on x.

In the BUGS language it is used as

`x ~ dloglik(lambda)`

##### Log-normal

The Log-normal distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dlnorm(mu, tau)`

##### Logistic

The Logistic distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dlogis(mu, tau)`

##### Normal

The Normal distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dnorm(mu, tau)`

##### Pareto

The Pareto distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dpar(alpha, c)`

##### Student-t

The Student-t distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dt(mu, tau, k)`

The Uniform distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dunif(a, b)`

##### Weibull

The Weibull distribution is defined by the pdf

In the BUGS language it is used as

`x ~ dweib(v, lambda)`

##### Multinomial

The Multinomial distribution is defined by the pmf

In the BUGS language it is used as

`x[] ~ dmulti(p[], N)`

##### Dirichlet

The Dirichlet distribution is defined by the pdf

In the BUGS language it is used as

`p[] ~ ddirich(alpha[])`

It may also be spelt

`ddirch`

as in WinBUGS.

##### Multivariate normal

The Multivariate Normal distribution is defined by the pdf

In the BUGS language it is used as

`x[] ~ dmnorm(mu[], T[,])`

##### Multivariate Student-t

The Multivariate Student-t distribution is defined by the pdf

In the BUGS language it is used as

`x[] ~ dmt(mu[], T[,], k)`

##### Wishart

The Wishart distribution is defined by the pdf

In the BUGS language it is used as

`x[,] ~ dwish(R[,], k)`