Let X have a t-distribution with r degrees of freedom.
How would I go about showing:
E(X) = 0 , r > 1
and
Var(X) = r/(r-2) , r > 2
Here is a recipe which I think will allow you to find with a minimum of effort.
1. Show that E(X) = 0, so .
2. Show that .
3. Write down the integral for . Make the substitution . Notice that the resulting integral is of the form , where c is a constant (depending on r, but independent of t), and f is the pdf of a t distribution with r-2 degrees of freedom.
4. Use the fact that for any pdf f(t) to evaluate the integral in 3.
5. Substitute into the equation in 2. and solve for .
Like Mr. Fantastic, I find this question confusing. You don't need to know any of those things, provided you know the formula for the pdf of a t distribution. If you don't know the pdf, then that's a different problem.
One thing I forgot to mention in my original post: you will need the identity to get the answer in its final form.