2) "If X is non-negative, then E(X) = Integral(0 to infinity) of (1-F(x))dx, where F(x) is the cumulative distribution function of X."
[Aside: the source that I quote this from says that the above is true no matter X is discrete or continuous. But if X is discrete, how can E(X) have an integral in it? It doesn't make much intuive sense to me...]
I tried integration by parts (letting u = 1 - F(x), dv=dx) and I think I am done if I can prove that
lim x(1-F(x)) = 0
But this actually gives "infinity times 0" which is an indeterminate form and requires L'Hopital's Rule. I tried many different ways but was still unable to figure out what the limit is going to be...how can we prove that the limit is equal to 0?