Chi-squared Distribution

Definition

$X\sim {\chi }^2(k)=\Gamma \left(\frac{k}{2},2\right)$ where $k\in \mathbb{N}$ is the degrees of freedom

squared sum of independent standard normal distributions

Properties

PDF

$f(x)=\frac{1}{\Gamma (\frac{k}{2})2^{\frac{k}{2}}}{x}^{\frac{k}{2}-1}{e}^{-\frac{x}{2}}\in {\mathbb{R}}_0^{+}$

Mean

$E(X)=k$

Variance

$\mathrm{Var}(X)=2k$

MGF

$M(t)=(1-2t{)}^{-k/2}$

Additivity

Let $X_i\sim {\chi }^2(k_i)$‘s are independent chi-squared distributions $\sum _{i=1}^n{X}_i\sim {\chi }^2\left(\sum _{i=1}^n{k}_i\right)$

Facts

Let $Q_1\sim {\chi }^2(r_1)$ and $Q_2\sim {\chi }^2(r_2)$, and $Q=Q_1-Q_2$ is independent of $Q_2$, then $Q\sim {\chi }^2(r_1-r_2)$

Let $Q=Q_1+Q_2+\cdots +Q_{k-1}+Q_k$, where $Q,Q_1,Q_2,\dots ,Q_k$ are quadratic forms in $\mathbf{x}$, where each element of the $\mathbf{x}$ is a Random Sample from $N(\mu ,{\sigma }^2)$ If $Q/{\sigma }^2\sim {\chi }^2(r),Q_1/{\sigma }^2\sim {\chi }^2(r_1),Q_2/{\sigma }^2\sim {\chi }^2(r_2),\dots ,Q_{k-1}/{\sigma }^2\sim {\chi }^2(r_{k-1})$, then

• $Q_1,Q_2,\dots ,Q_k$ are independent
• $Q_k/{\sigma }^2\sim {\chi }^2(r-r_1-r_2-\cdots -r_{k-1})$