Can you help me with this:
Let Y be a random variable taking values 0 and 1 with probability 0.5
each. Find expected values of random variable Z = X·Y
I think the issue might be this: The problem asks you to find the expected values of Z and their probabilities, not the expected value of Z. This is probably a poor choice of words. It would be better to call them possible values or something like that.
It is clear that the possible values of Z are the same as those of X, since multiplying with Y means multiplying by either 0 or 1. Can you find the new probabilities?