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## 13.5 Poisson Distribution

### 13.5.1 Probability Mass Function

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

### 13.5.2 Sampling Statement

n ~ poisson(lambda)

Increment target log probability density with poisson_lpmf(n | lambda) dropping constant additive terms.

### 13.5.3 Stan Functions

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

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.