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## 18.6 Gamma distribution

### 18.6.1 Probability density function

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

### 18.6.2 Sampling statement

y ~ gamma(alpha, beta)

Increment target log probability density with gamma_lupdf(y | alpha, beta).
Available since 2.0

### 18.6.3 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
Available since 2.12

real gamma_lupdf(reals y | reals alpha, reals beta)
The log of the gamma density of y given shape alpha and inverse scale beta dropping constant additive terms
Available since 2.25

real gamma_cdf(reals y, reals alpha, reals beta)
The cumulative gamma distribution function of y given shape alpha and inverse scale beta
Available since 2.0

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
Available since 2.12

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
Available since 2.12

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.
Available since 2.18