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## 16.2 Chi-Square Distribution

### 16.2.1 Probability Density Function

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

### 16.2.2 Sampling Statement

y ~ chi_square(nu)

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

### 16.2.3 Stan Functions

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

real chi_square_cdf(reals y, reals nu)
The Chi-square cumulative distribution function of y given degrees of freedom nu

real chi_square_lcdf(reals y | reals nu)
The log of the Chi-square cumulative distribution function of y given degrees of freedom nu

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

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