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## 17.3 Inverse Chi-Square Distribution

### 17.3.1 Probability Density Function

If $$\nu \in \mathbb{R}^+$$, then for $$y \in \mathbb{R}^+$$, $\text{InvChiSquare}(y \, | \, \nu) = \frac{2^{-\nu/2}} {\Gamma(\nu / 2)} \, y^{-\nu/2 - 1} \, \exp\! \left( \! - \, \frac{1}{2} \, \frac{1}{y} \right) .$

### 17.3.2 Sampling Statement

y ~ inv_chi_square(nu)

Increment target log probability density with inv_chi_square_lupdf(y | nu).

### 17.3.3 Stan Functions

real inv_chi_square_lpdf(reals y | reals nu)
The log of the inverse Chi-square density of y given degrees of freedom nu

real inv_chi_square_lupdf(reals y | reals nu)
The log of the inverse Chi-square density of y given degrees of freedom nu dropping constant additive terms

real inv_chi_square_cdf(reals y, reals nu)
The inverse Chi-squared cumulative distribution function of y given degrees of freedom nu

real inv_chi_square_lcdf(reals y | reals nu)
The log of the inverse Chi-squared cumulative distribution function of y given degrees of freedom nu

real inv_chi_square_lccdf(reals y | reals nu)
The log of the inverse Chi-squared complementary cumulative distribution function of y given degrees of freedom nu

R inv_chi_square_rng(reals nu)
Generate an inverse Chi-squared variate with degrees of freedom nu; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.