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19.8 Weibull distribution

19.8.1 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 $Y^{-1} \propto \text{Frechet}(\alpha,\sigma^{-1})$.

19.8.2 Sampling statement

y ~ weibull(alpha, sigma)

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

19.8.3 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
Available since 2.12

real weibull_lupdf(reals y | reals alpha, reals sigma)
The log of the Weibull density of y given shape alpha and scale sigma dropping constant additive terms
Available since 2.25

real weibull_cdf(reals y, reals alpha, reals sigma)
The Weibull cumulative distribution function of y given shape alpha and scale sigma
Available since 2.0

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
Available since 2.12

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
Available since 2.12

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.
Available since 2.18