Question

Assume X1, . . . , Xn iid normal with mean $\mu$ and variance $\sigma$ ^2 , show that

a. X¯ and X^2 are independent.

b. Proof that X¯ is normally distributed with mean $\mu$ and variance $\sigma$ ^2/n.

c. Proof that (n ? 1)S^2/?2 is chi-squared distributed with (n ? 1) degrees of freedom.

d. Show that X¯ S/ $\sqrt{n}$ is t distributed with (n ? 1) degrees of freedom

Answer 1

$The ~joint~distribution~X_1,X_2,...,X_n~is\\ \frac{1}{\left ( \sigma\sqrt{2\pi} \right )^n}\exp\left [ -\frac{1}{2\sigma^2}\sum_{i=1}^n(x_i-\mu)^2 \right ]~with~-\infty<x_i<\infty~\forall~i=1,2,...,n\\ Transform~(X_1,X_2,...,X_n)\rightarrow (Y_1,Y_2,...,Y_n)~such~that\\ y_1=\frac{1}{\sqrt{n}}(x_1-\mu)+\frac{1}{\sqrt{n}}(x_2-\mu)+...+\frac{1}{\sqrt{n}}(x_n-\mu)\\ y_2=\frac{1}{\sqrt{6}}(x_1-\mu)+\frac{1}{\sqrt{6}}(x_2-\mu)-\frac{2}{\sqrt{6}}(x_3-\mu)\\ .\\ .\\ .\\ y_n=\frac{1}{\sqrt{n(n-1)}}(x_1-\mu)+\frac{1}{\sqrt{n(n-1)}}(x_2-\mu)+...+\frac{1}{\sqrt{n(n-1)}}(x_{n-1}-\mu)-\frac{n-1}{\sqrt{n(n-1)}}(x_n-\mu)$

This transformation is called Helmert's transformation (orthogonal transformation).

Under this orthogonal transformation,

$\sum_{i=1}^ny_i^2=\sum_{i=1}^n(x_i-\mu)^2\\ Jacobian, J \left ( \frac{x_1,...,x_n}{y_1,...,y_n} \right )=\pm 1\\ Hence ~the ~joint~p.d.f.~of~Y_1,Y_2,...,Y_n~is\\ \frac{1}{(\sigma\sqrt{2\pi})^n}\exp\left [ -\frac{1}{2\sigma^2}\sum_{i=1}^ny_i^2 \right ]~~with~-\infty<y_i<\infty,~i=1,2,...,n\\ This~shows~that~Y_i~(i=1,2,...,n)~are~mutually~independent~random~variables~each~distributed~as~ a~normal~variable~with~mean~0~and~variance~\sigma^2.\\$ $(a)Y_1=\frac{1}{\sqrt{n}}\sum_{i=1}^n(X_i-\mu)=\sqrt{n}(\bar{X}-\mu)\\ Or~\bar{X}=\mu+\frac{1}{\sqrt{n}}Y_1\\ ~\sum_{i=1}^nY_i^2=\sum_{i=1}^n(X_i-\mu)^2\\ Y_1^2=n(\bar{X}-\mu)^2\\ \sum_{i=2}^nY_i^2=\sum_{i=1}^n(X_i-\mu)^2-n(\bar{X}-\mu)^2=\sum_{i=1}^n(X_I-\bar{X})^2=(n-1)S^2\\ Since~Y_i's~are~independent~then~bar{X}~and~S^2~are~independent.\\ (b) Y_1=\frac{1}{\sqrt{n}}\sum_{i=1}^n(X_i-\mu)=\sqrt{n}(\bar{X}-\mu)\\ Or~\bar{X}=\mu+\frac{1}{\sqrt{n}}Y_1\sim N(\mu,~\frac{\sigma^2}{n}),~since~Y_1\sim N(0,~\sigma^2)\\ (c) ~since ~Y_i's~are ~independently~N(0,~\sigma^2) ~or~\frac{Y_i}{\sigma}'s~are ~independently~N(0,~1)\\ then~\sum_{i=2}^n\frac{Y_i^2}{\sigma^2}=(n-1)S^2/\sigma^2\sim \chi^2_{n-1}\\ d)~\frac{\bar{X}-\mu}{\sigma}\sim N(0,1)\\ (n-1)S^2/\sigma^2\sim \chi^2_{n-1}\\ and~they~are~independent~hence\\ \frac{\frac{\sqrt{n}(\bar{X}-\mu)}{\sigma}}{\sqrt{\frac{(n-1)S^2}{(n-1)\sigma^2}}}=\frac{\sqrt{n}(\bar{X}-\mu)}{S}\sim t_{n-1}$