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 transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.