Definition:Chi Distribution

Definition
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\Img X = \hointr 0 \infty$.

Let $r$ be a strictly positive integer.

$X$ is said to have a chi distribution with $r$ degrees of freedom if it has probability density function:


 * $\displaystyle \map {f_X} x = \dfrac 1 {2^{\paren {r / 2} - 1} \map \Gamma {r / 2} } x^{r - 1} e^{- x^2 / 2}$

where $\Gamma$ is the gamma function.

This is written:


 * $X \sim \chi_r$