Chi Squared Distribution

A ${\chi }^2$ Distribution is a kind of Sampling Distribution formed by combining iid squared Z Distributions through addition.

• ${\chi }^2$ shows up when we try to estimate Variance

$$\frac{Z_{\mathrm{1..}k}{\sim }^{\text{iid}}N(0,1)}{{\chi }_{(k)}^2\sim {\sum }_{i=1}^k{Z}_i^2}$$

• $E[Y]=k$
• $\text{Var}(Y)=2k$
• ${\chi }_{(1)}^2=Z^2$
• ${\chi }_{(2)}^2=\text{Exp}\left(\frac{1}{2}\right)$
• ${\chi }_{(\text{big})}^2\approx N$
• $k$ is Degree of Freedom

Additive Property

The sum of iid ${\chi }^2$ variables is also ${\chi }^2$

$$\frac{Y_{\mathrm{1..}n}{\sim }^{\text{indep}}{\chi }_{(k_i)}^2}{{\sum }_{i=1}^n{Y}_i\sim {\chi }_{\left({\sum }_{i=1}^n{k}_i\right)}^2}$$

Chi Squared Quantiles

If $Y\sim {\chi }_{(k)}^2$ then $(1-\alpha )$ quantile is denoted ${\chi }_{\alpha ,k}^2$

• where $P(Y>{\chi }_{\alpha ,k}^2)=1-F({\chi }_{\alpha ,k}^2)=\alpha$
• where $F$ is the ${\chi }^2$ CDF
• where $\alpha$ is type I error
• where $k$ is Degree of Freedom