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## 19.2 Chi-square distribution

### 19.2.1 Probability density function

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

### 19.2.2 Sampling statement

y ~ chi_square(nu)

Increment target log probability density with chi_square_lupdf(y | nu).
Available since 2.0

### 19.2.3 Stan functions

real chi_square_lpdf(reals y | reals nu)
The log of the Chi-square density of y given degrees of freedom nu
Available since 2.12

real chi_square_lupdf(reals y | reals nu)
The log of the Chi-square density of y given degrees of freedom nu dropping constant additive terms
Available since 2.25

real chi_square_cdf(reals y, reals nu)
The Chi-square cumulative distribution function of y given degrees of freedom nu
Available since 2.0

real chi_square_lcdf(reals y | reals nu)
The log of the Chi-square cumulative distribution function of y given degrees of freedom nu
Available since 2.12

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
Available since 2.12

R chi_square_rng(reals nu)
Generate a Chi-square 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.
Available since 2.18