Distributions arising from the normal distribution

Let $X_{1}$ and $X_{2}$ be independent variables with the $\chi^{2}(m)$ and $\chi^{2}(n)$ distributions respectively. Show that $X_{1}+X_{2}$ has the $\chi^{2}(m+n)$ distribution.

Show that the mean of the $t(r)$ distribution is $0$, and that the mean of the $F(r, s)$ distribution is $s /(s-2)$ if $s > 2$. What happens if $s \leq 2$?

Show that the $t(1)$ distribution and the Cauchy distribution are the same.

Let $X$ and $Y$ be independent variables having the exponential distribution with parameter 1. Show that $X / Y$ has an $F$ distribution. Which?

Show the independence of the sample mean and sample variance of an independent sample from the $N\left(\mu , \sigma^{2}\right)$ distribution. This may be done by either or both of: (i) the result of Exercise (4.5.7), (ii) induction on $n$.

Let $\left\{ X_{r}: 1 \leq r \leq n\right\}$ be independent $N(0,1)$ variables. Let $\Psi \in [0, \pi ]$ be the angle between the vector $\left(X_{1}, X_{2}, \ldots , X_{n}\right)$ and some fixed vector in $\mathbb {R}^{n}$. Show that $\Psi$ has density $f(\psi ) = (\sin \psi )^{n-2} / B\left(\frac{1}{2}, \frac{1}{2} n-\frac{1}{2}\right), 0 \leq \psi < \pi$, where $B$ is the beta function.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent $N(0,1)$ random variables, and let $V_{n} \geq 0$ be given by $V_{n}^{2} = \sum_{i = 1}^{n} X_{i}^{2}$. Show that the random vector $Y = \left(X_{1}, X_{2}, \ldots , X_{n}\right) / V_{n}$ is uniformly distributed on the unit sphere. Deduce the result of Exercise (4.10.6).