16.6 Gamma Distribution

16.6.1 Probability Density Function

If \(\alpha \in \mathbb{R}^+\) and \(\beta \in \mathbb{R}^+\), then for \(y \in \mathbb{R}^+\), \[ \text{Gamma}(y|\alpha,\beta) = \frac{\beta^{\alpha}} {\Gamma(\alpha)} \, y^{\alpha - 1} \exp(-\beta \, y) . \]

16.6.2 Sampling Statement

y ~ gamma(alpha, beta)

Increment target log probability density with gamma_lpdf( y | alpha, beta) dropping constant additive terms.

16.6.3 Stan Functions

real gamma_lpdf(reals y | reals alpha, reals beta)
The log of the gamma density of y given shape alpha and inverse scale beta

real gamma_cdf(reals y, reals alpha, reals beta)
The cumulative gamma distribution function of y given shape alpha and inverse scale beta

real gamma_lcdf(reals y | reals alpha, reals beta)
The log of the cumulative gamma distribution function of y given shape alpha and inverse scale beta

real gamma_lccdf(reals y | reals alpha, reals beta)
The log of the complementary cumulative gamma distribution function of y given shape alpha and inverse scale beta

R gamma_rng(reals alpha, reals beta)
Generate a gamma variate with shape alpha and inverse scale beta; may only be used in generated quantities block. For a description of argument and return types, see section vectorized PRNG functions.