Let xn = x2 1 + + x2 n, where the xi's are iid standard normal random variables.

(a) show that sn is a chi-square random variable with n de- grees of freedom.

hint: show that x2 i is chi-square with one degree of freedom, and then use problem 6.

(b) find the pdf of tn = (x2 1 + + x2 n)1=2

(c) show that t2 is a rayleigh random variable.

(d) find the pdf for t3. the random variable t3 is used to model the speed of molecules in a gas. it is said to have the maxwell distribution.