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14.5 Poisson Distribution

14.5.1 Probability Mass Function

If \(\lambda \in \mathbb{R}^+\), then for \(n \in \mathbb{N}\), \[ \text{Poisson}(n|\lambda) = \frac{1}{n!} \, \lambda^n \, \exp(-\lambda). \]

14.5.2 Sampling Statement

n ~ poisson(lambda)

Increment target log probability density with poisson_lupmf(n | lambda).

14.5.3 Stan Functions

real poisson_lpmf(ints n | reals lambda)
The log Poisson probability mass of n given rate lambda

real poisson_lupmf(ints n | reals lambda)
The log Poisson probability mass of n given rate lambda dropping constant additive terms

real poisson_cdf(ints n, reals lambda)
The Poisson cumulative distribution function of n given rate lambda

real poisson_lcdf(ints n | reals lambda)
The log of the Poisson cumulative distribution function of n given rate lambda

real poisson_lccdf(ints n | reals lambda)
The log of the Poisson complementary cumulative distribution function of n given rate lambda

R poisson_rng(reals lambda)
Generate a Poisson variate with rate lambda; may only be used in transformed data and generated quantities blocks. lambda must be less than \(2^{30}\). For a description of argument and return types, see section vectorized function signatures.