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## 17.7 Inverse Gamma Distribution

### 17.7.1 Probability Density Function

If $$\alpha \in \mathbb{R}^+$$ and $$\beta \in \mathbb{R}^+$$, then for $$y \in \mathbb{R}^+$$, $\text{InvGamma}(y|\alpha,\beta) = \frac{\beta^{\alpha}} {\Gamma(\alpha)} \ y^{-(\alpha + 1)} \, \exp \! \left( \! - \beta \, \frac{1}{y} \right) .$

### 17.7.2 Sampling Statement

y ~ inv_gamma(alpha, beta)

Increment target log probability density with inv_gamma_lupdf(y | alpha, beta).

### 17.7.3 Stan Functions

real inv_gamma_lpdf(reals y | reals alpha, reals beta)
The log of the inverse gamma density of y given shape alpha and scale beta

real inv_gamma_lupdf(reals y | reals alpha, reals beta)
The log of the inverse gamma density of y given shape alpha and scale beta dropping constant additive terms

real inv_gamma_cdf(reals y, reals alpha, reals beta)
The inverse gamma cumulative distribution function of y given shape alpha and scale beta

real inv_gamma_lcdf(reals y | reals alpha, reals beta)
The log of the inverse gamma cumulative distribution function of y given shape alpha and scale beta

real inv_gamma_lccdf(reals y | reals alpha, reals beta)
The log of the inverse gamma complementary cumulative distribution function of y given shape alpha and scale beta

R inv_gamma_rng(reals alpha, reals beta)
Generate an inverse gamma variate with shape alpha and scale beta; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.