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## 16.8 Logistic distribution

### 16.8.1 Probability density function

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

### 16.8.2 Sampling statement

y ~ logistic(mu, sigma)

Increment target log probability density with logistic_lupdf(y | mu, sigma).

### 16.8.3 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_lupdf(reals y | reals mu, reals sigma)
The log of the logistic density of y given location mu and scale sigma dropping constant additive terms

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.