I am working on the following problem:
Given two discrete random variables, X and Y - which each take one of two values with probability 0.5 - are they uncorrelated?
I'm not permitted to assume independence, because that is a later question. I know that $Cov(x,y) = 0 \Rightarrow E[XY] -\mu_x\mu_y = 0$. I can find the means quite easily. But I can't figure out how to calculate the expectation of $XY$, given that I don't know what I can assume about the probability of their ordered pairs.
A hint on how I can make progress on calculating $E[XY]$ would be appreciated.