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## 15.3 Exponentially Modified Normal Distribution

### 15.3.1 Probability Density Function

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

### 15.3.2 Sampling Statement

y ~ exp_mod_normal(mu, sigma, lambda)

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

### 15.3.3 Stan Functions

real exp_mod_normal_lpdf(reals y | reals mu, reals sigma, reals lambda)
The log of the exponentially modified normal density of y given location mu, scale sigma, and shape lambda

real exp_mod_normal_cdf(reals y, reals mu, reals sigma, reals lambda)
The exponentially modified normal cumulative distribution function of y given location mu, scale sigma, and shape lambda

real exp_mod_normal_lcdf(reals y | reals mu, reals sigma, reals lambda)
The log of the exponentially modified normal cumulative distribution function of y given location mu, scale sigma, and shape lambda

real exp_mod_normal_lccdf(reals y | reals mu, reals sigma, reals lambda)
The log of the exponentially modified normal complementary cumulative distribution function of y given location mu, scale sigma, and shape lambda

R exp_mod_normal_rng(reals mu, reals sigma, reals lambda)
Generate a exponentially modified normal variate with location mu, scale sigma, and shape lambda; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.