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