## 16.1 Lognormal Distribution

### 16.1.1 Probability Density Function

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

### 16.1.2 Sampling Statement

y ~ lognormal(mu, sigma)

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

### 16.1.3 Stan Functions

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

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

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

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

R lognormal_rng(reals mu, reals sigma)
Generate a lognormal 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.