Probability and Statistics: Let X and Y be independent random variables, and E(A) be the expectation of any random variable A. Simplify this expression: E(2XY - 2XE(Y) - 2YE(X) + 2E(X)E(Y))

I think it's supposed to be 0, by the way, but I just don't know how this turns out to be 0.

Question

Probability and Statistics: Let X and Y be independent random variables, and E(A) be the expectation of any random variable A. Simplify this expression: E(2XY - 2XE(Y) - 2YE(X) + 2E(X)E(Y))

I think it's supposed to be 0, by the way, but I just don't know how this turns out to be 0.

anonymous · Answer

*please note the correction: E(2XY... instead of E(4XY...

anonymous · Answer

Do you know about Covariance?  Somehow feel this plays a part.

anonymous · Answer

yes, but how..?

anonymous · Answer

Well, what formula do you know about covariance that involve the expected value and independent random variables?

anonymous · Answer

Thank you, I just figured out the solution (through looking deeper into covariance). This question is actually part of a larger proof that I'm trying to do, which is the variance of the sum of two independent random variables. I attached the proof, please review it if you wish, for correctness.