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## 15.9 Gumbel Distribution

### 15.9.1 Probability Density Function

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

### 15.9.2 Sampling Statement

y ~ gumbel(mu, beta)

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

### 15.9.3 Stan Functions

real gumbel_lpdf(reals y | reals mu, reals beta)
The log of the gumbel density of y given location mu and scale beta

real gumbel_cdf(reals y, reals mu, reals beta)
The gumbel cumulative distribution function of y given location mu and scale beta

real gumbel_lcdf(reals y | reals mu, reals beta)
The log of the gumbel cumulative distribution function of y given location mu and scale beta

real gumbel_lccdf(reals y | reals mu, reals beta)
The log of the gumbel complementary cumulative distribution function of y given location mu and scale beta

R gumbel_rng(reals mu, reals beta)
Generate a gumbel variate with location mu and 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.