Let Y be random variable with gamma distribution with parameters a and ß. Let a be a number with a + a > 0. Find out a simple expression for E(Ya). (Hints: You shouldn't have to do any integration; instead you must transform the thing you've to integrate in a different gamma density function. Your final answer must be of the form (constant) G (something)/ G (something else).)
Let Y ~ ?2(?). Find E(1/Y).
Employ your answer above to find out the variance of t-distribution. (Hint: If Z is standard normal, then you know the distribution of Z2.)
Take the limit of variance above as ? Goes to infinity. (This should be very easy if you did everything right so far.)
But of course you already knew what that limit above was going to be before you started this problem. Why?