Chi-Squared Distribution

A Continuous Probability Distributions ${\mathcal{X}}^2$ for Genetics.

• Let $U=Z^2$ where $Z$ is a Standard Normal Variable
• $U{\sum }_{i=1}^n{\mathcal{X}}^2$ distribution with 1 degree of freedom written as $X_{(1)}^2$

Definition

With $Z_1,\dots ,Z_n{\sim }_{iid}N(0,1)$, then:

$$\{\begin{array}{l}{Z}_1^2\sim \Gamma (\frac{1}{2},\frac{1}{2})={\mathcal{X}}^2(1)\\ Z_1^2+\cdots +Z_n^2\sim \Gamma (\frac{1}{2},\frac{1}{2})={\mathcal{X}}^2(n)\end{array}$$

Sampling Distribution

$$\frac{(n-1)S_n^2}{{\sigma }^2}=\frac{{\sum }_{i=1}^n(X_i-{\overline{X}}_n{)}^2}{{\sigma }^2}\sim {\mathcal{X}}^2(n-1)$$

Properties

• Additive property: $X\sim {\mathcal{X}}_{(m)}^2$, $X\perp Y⟹X+Y\sim {\mathcal{X}}_{(m+n)}^2$
  • If ${\mathcal{X}}_{(m)}^2\equiv X_{(1)}^2+\cdots +X_{(1)}^2=Z_1^2+\cdots +Z_{(m)}^2$
• If $X\sim X_{(m)}^2⟹E[X]=m$