Question

The density function of a chi-squared random variable having n degrees of freedom can be shown to be f(x) = Je-*/2(x/237–1 T(n/2) , X > 0 where I(t) is the gamma function defined by r(t) = ( e*xl-1dx, t> 0 Integration by parts can be employed to show that r(t) = (t-1)F(t-1), when t > 1. If Z and x7 are independent random variables with Z having a standard normal distribution and having a chi-square distribution with n degrees of freedom, then the random variable T defined by Z T = x/n is said to have a t-distribution with n degrees of freedom. Compute its mean and variance when n > 2.