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## 16.8 Weibull Distribution

### 16.8.1 Probability Density Function

If $$\alpha \in \mathbb{R}^+$$ and $$\sigma \in \mathbb{R}^+$$, then for $$y \in [0,\infty)$$, $\text{Weibull}(y|\alpha,\sigma) = \frac{\alpha}{\sigma} \, \left( \frac{y}{\sigma} \right)^{\alpha - 1} \, \exp \! \left( \! - \left( \frac{y}{\sigma} \right)^{\alpha} \right) .$

Note that if $$Y \propto \text{Weibull}(\alpha,\sigma)$$, then $$Y^{-1} \propto \text{Frechet}(\alpha,\sigma^{-1})$$.

### 16.8.2 Sampling Statement

y ~ weibull(alpha, sigma)

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

### 16.8.3 Stan Functions

real weibull_lpdf(reals y | reals alpha, reals sigma)
The log of the Weibull density of y given shape alpha and scale sigma

real weibull_cdf(reals y, reals alpha, reals sigma)
The Weibull cumulative distribution function of y given shape alpha and scale sigma

real weibull_lcdf(reals y | reals alpha, reals sigma)
The log of the Weibull cumulative distribution function of y given shape alpha and scale sigma

real weibull_lccdf(reals y | reals alpha, reals sigma)
The log of the Weibull complementary cumulative distribution function of y given shape alpha and scale sigma

R weibull_rng(reals alpha, reals sigma)
Generate a weibull variate with shape alpha and scale sigma; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.