Chi-Squared Distribution

probability-theory statistics

Definition

${\chi }^2$ Distribution

The ${\chi }_n^2$ distribution with $n$ degrees of freedom is the sum of $n$ independent squared standard normal random variables.

Let $X_i\stackrel{\text{ind}}{\sim }\mathcal{N}(0,1)$ for $i=1,\dots ,n$, then:

$$Y=\sum _{i=1}^n{X}_i^2\sim {\chi }_k^2$$

Probability Density Function

The density $f$ of $X\sim {\chi }_k^2$ random variable is:

$$f(x)=\frac{x^{n/2-1}{e}^{-x/2}}{2^{n/2}\Gamma (n/2)}$$

for $x>0$ and zero else and $\Gamma$ being the gamma function