(Proof of the independence of $\overline{\mathrm{X}}$ and $\mathrm{S}^{2}$ ) If $X_{1}, X_{2}, \ldots, X_{n}$ constitute a random sample from a normal population with the mean $\mu$ and the variance $\sigma^{2}$.
(a) find the conditional density of $X_{1}$ given $X_{2}=x_{2}, X_{3}=$ $x_{3}, \ldots, X_{n}=x_{n}$, and then set $X_{1}=n \bar{X}-X_{2}-\cdots-X_{n}$
and use the transformation technique to find the conditional density of $\bar{X}$ given $X_{2}=x_{2}, X_{3}=x_{3}, \ldots, X_{n}=x_{n}$
(b) find the joint density of $\bar{X}, X_{2}, X_{3}, \ldots, X_{n}$ by multiplying the conditional density of $\bar{X}$ obtained in part (a) by the joint density of $X_{2}, X_{3}, \ldots, X_{n}$, and show that
$$
g\left(x_{2}, x_{3}, \ldots, x_{n} \mid \bar{x}\right)=\sqrt{n}\left(\frac{1}{\sigma \sqrt{2 \pi}}\right)^{n-1} e^{-\frac{\omega-1 j^{2}}{2 \sigma^{2}}}
$$
for $-\infty<x_{i}<\infty, i=2,3, \ldots, n ;$
(c) show that the conditional moment-generating function of $\frac{(n-1) S^{2}}{\sigma^{2}}$ given $\bar{X}=\bar{x}$ is
$$
E\left[e^{\frac{(n-1) S^{2}}{\sigma^{2}} \cdot t} \mid \bar{x}\right]=(1-2 t)^{-\frac{n-1}{2}} \quad \text { for } t<\frac{1}{2}
$$
Since this result is free of $\bar{x}$, it follows that $\bar{X}$ and $S^{2}$ are independent; it also shows that $\frac{(n-1) S^{2}}{\sigma^{2}}$ has a chi-square distribution with $n-1$ degrees of freedom.
This proof, due to $\mathrm{J}$. Shuster, is listed among the references at the end of this chapter.