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## 15.8 Logistic Distribution

### 15.8.1 Probability Density Function

If $$\mu \in \mathbb{R}$$ and $$\sigma \in \mathbb{R}^+$$, then for $$y \in \mathbb{R}$$, $\text{Logistic}(y|\mu,\sigma) = \frac{1}{\sigma} \ \exp\!\left( - \, \frac{y - \mu}{\sigma} \right) \ \left(1 + \exp \!\left( - \, \frac{y - \mu}{\sigma} \right) \right)^{\!-2} \! .$

### 15.8.2 Sampling Statement

y ~ logistic(mu, sigma)

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

### 15.8.3 Stan Functions

real logistic_lpdf(reals y | reals mu, reals sigma)
The log of the logistic density of y given location mu and scale sigma

real logistic_cdf(reals y, reals mu, reals sigma)
The logistic cumulative distribution function of y given location mu and scale sigma

real logistic_lcdf(reals y | reals mu, reals sigma)
The log of the logistic cumulative distribution function of y given location mu and scale sigma

real logistic_lccdf(reals y | reals mu, reals sigma)
The log of the logistic complementary cumulative distribution function of y given location mu and scale sigma

R logistic_rng(reals mu, reals sigma)
Generate a logistic variate with location mu 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.