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## 16.4 Scaled Inverse Chi-Square Distribution

### 16.4.1 Probability Density Function

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

### 16.4.2 Sampling Statement

y ~ scaled_inv_chi_square(nu, sigma)

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

### 16.4.3 Stan Functions

real scaled_inv_chi_square_lpdf(reals y | reals nu, reals sigma)
The log of the scaled inverse Chi-square density of y given degrees of freedom nu and scale sigma

real scaled_inv_chi_square_cdf(reals y, reals nu, reals sigma)
The scaled inverse Chi-square cumulative distribution function of y given degrees of freedom nu and scale sigma

real scaled_inv_chi_square_lcdf(reals y | reals nu, reals sigma)
The log of the scaled inverse Chi-square cumulative distribution function of y given degrees of freedom nu and scale sigma

real scaled_inv_chi_square_lccdf(reals y | reals nu, reals sigma)
The log of the scaled inverse Chi-square complementary cumulative distribution function of y given degrees of freedom nu and scale sigma

R scaled_inv_chi_square_rng(reals nu, reals sigma)
Generate a scaled inverse Chi-squared variate with degrees of freedom nu and scale sigma; may only be used in generated quantities block. For a description of argument and return types, see section vectorized PRNG functions.