Random Variables and Expected Values X and Y are random variables with identical distributions. How do you prove/disprove X = Y and E(X) = E(Y). I am not looking for a ready-made answer. I would appreciate it if someone would point me in the right direction towards the appropriate resource.

Look up the Kolmorgov-Smirnov test. Use one of your datasets (X or Y) as the "reference distribution" and the other as the "test" distribution and see if you get a significant result..if so, then they are different.

Also note that if X and Y have the same distribution, then that implies that E(X)=E(Y), so this covers tha t issues as well. To test just the equality of means, you can use a number of "tests of location" for this purpose. This link has a nice list.