Chi-Squared Distribution

Definition

If $Z_1,\dots ,Z_k$ are independent, standard normal random variables, then the sum of their squares

$$Q=\sum _{i=1}^k{Z}_i^2$$

is distributed according to the chi-squared distribution with $k$ degrees of freedom. This is usually denoted as

$$Q\sim {\chi }^2(k)\text{or}Q\sim {\chi }_k^2$$

Probability density function

The PDF of the chi-squared distribution is

$$f(x;k)=\{\begin{array}{ll}\frac{x^{k/2-1}{e}^{-x/2}}{2^{k/2}\Gamma \left(\frac{k}{2}\right)},& x>0\\ 0,& \text{otherwise}\end{array}$$

where $\Gamma (k/2)$ denotes the gamma function

Properties

Mean and Variance

• Mean: $k$
• Variance: $2k$

Cochran's theorem

If $Z_1,\dots ,Z_n$ are independent identically distributed (i.i.d.), standard normal random variables, then

$$\sum _{t=1}^n(Z_t-\bar{Z}{)}^2\sim {\chi }_{n-1}^2$$

where $\bar{Z}=\frac{1}{n}\sum _{t=1}^n{Z}_t$

Additivity

If $Y_1\sim {\chi }_n^2,Y_2\sim {\chi }_m^2$ and $Y_1,Y_2$ are independent. Then we have $Y=Y_1+Y_2\sim {\chi }_{m+n}^2$

Related distributions

If $X\sim {\chi }_2^2$ then $X\sim \mathrm{Expo}(1/2)$ is an Exponential Distribution. To generalize, if $X_1,\dots ,X_n$ i.i.d. $\sim \mathrm{Expo}(\lambda )$ and $\bar{X}=\frac{1}{n}\sum _{i=1}^n{X}_i$, then $2n\lambda \bar{X}\sim {\chi }_{2n}^2$

If $X\sim \mathrm{Unif}(0,1)$(Uniform Distributions), then $-2\ln (X)\sim {\chi }_2^2$. To generalize, if $Y_1,\dots ,Y_n$ i.i.d. $\sim \mathrm{Unif}(0,1)$, then $-2\sum _{i=1}^n\ln Y_i\sim {\chi }_{2n}^2$

${\chi }_n^2=\mathrm{Gamma}(n/2,1/2)$ (Gamma Distribution)

As $k\to \infty$, $\frac{1}{\sqrt{2k}}\left({\chi }_k^2-k\right)\stackrel{d}{\to }\mathcal{N}(0,1)$. (convergence in distribution)