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## 16.4 Skew Normal Distribution

### 16.4.1 Probability Density Function

If $$\xi \in \mathbb{R}$$, $$\omega \in \mathbb{R}^+$$, and $$\alpha \in \mathbb{R}$$, then for $$y \in \mathbb{R}$$, $\text{SkewNormal}(y \mid \xi, \omega, \alpha) = \frac{1}{\omega\sqrt{2\pi}} \ \exp\left( - \, \frac{1}{2} \left( \frac{y - \xi}{\omega} \right)^2 \right) \ \left(1 + \text{erf}\left( \alpha\left(\frac{y - \xi}{\omega\sqrt{2}}\right)\right)\right) .$

### 16.4.2 Sampling Statement

y ~ skew_normal(xi, omega, alpha)

Increment target log probability density with skew_normal_lupdf(y | xi, omega, alpha).

### 16.4.3 Stan Functions

real skew_normal_lpdf(reals y | reals xi, reals omega, reals alpha)
The log of the skew normal density of y given location xi, scale omega, and shape alpha

real skew_normal_lupdf(reals y | reals xi, reals omega, reals alpha)
The log of the skew normal density of y given location xi, scale omega, and shape alpha dropping constant additive terms

real skew_normal_cdf(reals y, reals xi, reals omega, reals alpha)
The skew normal distribution function of y given location xi, scale omega, and shape alpha

real skew_normal_lcdf(reals y | reals xi, reals omega, reals alpha)
The log of the skew normal cumulative distribution function of y given location xi, scale omega, and shape alpha

real skew_normal_lccdf(reals y | reals xi, reals omega, reals alpha)
The log of the skew normal complementary cumulative distribution function of y given location xi, scale omega, and shape alpha

R skew_normal_rng(reals xi, reals omega, real alpha)
Generate a skew normal variate with location xi, scale omega, and shape alpha; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.