chi

Let $gamma in mathbb{N} $. A continuous random variable $left(chi_{gamma}^{2}, f_{X}right)$ with density

$$f_{gamma}(x)=left{begin{array}{ll}{frac{1}{Gamma(gamma / 2) 2^{alpha}} x^{gamma / 2-1} e^{-x / 2},} & {x>0} \ {0,} & {x leq 0}end{array}right.$$

is said to follow a chi-squared distribution with $gamma$ degrees of freedom.

the chi-squared distribution is simply that of a gamma random variable with $beta$= 2 and $alpha$ =
=$gamma /2 $, hence, we see that

$$Eleft[chi_{gamma}^{2}right]=gamma, quad quad operatorname{Var}left[chi_{gamma}^{2}right]=2 gamma$$