Let X1,…,XnX_1, \dots, X_nX1,…,Xn be i.i.d. N(0,σ2)N(0, \sigma^2)N(0,σ2). What is the distribution of the sample variance S2=1n−1∑i=1n(Xi−Xˉ)2S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2S2=n−11∑i=1n(Xi−Xˉ)2?
(n−1)S2σ2∼χn2\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n}σ2(n−1)S2∼χn2
(n−1)S2σ2∼χn−12\frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1}σ2(n−1)S2∼χn−12
nS2σ2∼χn−12\frac{nS^2}{\sigma^2} \sim \chi^2_{n-1}σ2nS2∼χn−12
S2σ2∼χn−12\frac{S^2}{\sigma^2} \sim \chi^2_{n-1}σ2S2∼χn−12