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16.5 Poisson distribution

16.5.1 Probability mass function

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

16.5.2 Sampling statement

n ~ poisson(lambda)

Increment target log probability density with poisson_lupmf(n | lambda).
Available since 2.0

16.5.3 Stan functions

real poisson_lpmf(ints n | reals lambda)
The log Poisson probability mass of n given rate lambda
Available since 2.12

real poisson_lupmf(ints n | reals lambda)
The log Poisson probability mass of n given rate lambda dropping constant additive terms
Available since 2.25

real poisson_cdf(ints n, reals lambda)
The Poisson cumulative distribution function of n given rate lambda
Available since 2.0

real poisson_lcdf(ints n | reals lambda)
The log of the Poisson cumulative distribution function of n given rate lambda
Available since 2.12

real poisson_lccdf(ints n | reals lambda)
The log of the Poisson complementary cumulative distribution function of n given rate lambda
Available since 2.12

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.
Available since 2.18