Chi Squared Variance Independent Distribution

For $\frac{(n-1)S^2}{{\sigma }^2}\sim {\mathcal{X}}_{df=n-1}^2$

Proof

• Note that $\frac{{\sum }_{i=1}^n(x_i-\mu {)}^2}{{\sigma }^2}=\frac{{\sum }_{i=1}^n(x_i-\overline{x}{)}^2}{{\sigma }^2}+\frac{n(\overline{x}-\mu {)}^2}{{\sigma }^2}$

Examples

• With $V[{\chi }_{(m)}^2]=2m$
• Then, $V[\frac{(n-1)S^2}{{\sigma }^2}]=2(n-1)$
• $⟹V[S^2]\frac{(n-1{)}^2}{{\sigma }^4}=2(n-1)]$
• $⟹V[S^2]=\frac{2{\sigma }^4}{n-1}$