Question:
Let's say I have $X \sim N(\mu_1, \sigma) $ and $Y \sim N(\mu_2, \sigma) $. I know that $ cor(X,Y) = \rho $. What is $E(XY)$?
What I've tried
Based on a similar question where X and Y are independent, you might say:
$$XY = (X+Y)^2/4 - (X-Y)^2/4$$
The author of the answer to this post argues that since there is zero correlation between (X+Y) and (X-Y), and since $(X+Y)^2/4$ is a Cauchy R.V. (square of a normal), we can use convoutions to find the joint PDF from which we can find the expectation. I'm not sure that the zero correlation implies independence, so I'm not sure if you can do this and if all this holds when $X$ and $Y$ are dependent, too.
Thanks for your time!
Hint: If you know the correlation coefficient, then you can quickly compute the covariance, which is $E(XY)-E(X)E(Y)$.