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## 16.10 Rayleigh Distribution

### 16.10.1 Probability Density Function

If $$\sigma \in \mathbb{R}^+$$, then for $$y \in [0,\infty)$$, $\text{Rayleigh}(y|\sigma) = \frac{y}{\sigma^2} \exp(-y^2 / 2\sigma^2) \!.$

### 16.10.2 Sampling Statement

y ~ rayleigh(sigma)

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

### 16.10.3 Stan Functions

real rayleigh_lpdf(reals y | reals sigma)
The log of the Rayleigh density of y given scale sigma

real rayleigh_cdf(real y, real sigma)
The Rayleigh cumulative distribution of y given scale sigma

real rayleigh_lcdf(real y | real sigma)
The log of the Rayleigh cumulative distribution of y given scale sigma

real rayleigh_lccdf(real y | real sigma)
The log of the Rayleigh complementary cumulative distribution of y given scale sigma

R rayleigh_rng(reals sigma)
Generate a Rayleigh variate with scale sigma; may only be used in generated quantities block. For a description of argument and return types, see section vectorized PRNG functions.