I have 3 independent Normal Random Variables: $A$, $B$ and $C$, each with mean=$0$ and Variance $1$.
Then I have $X=3A+5B$ and $Y=A-C$... because both of them are functions of $A$, we know they are not independent.
I have calculated the means and variance of both $X$ and $Y$, but now I need to calculate the $E[XY]$ and not sure how to approach the problem.
I'm inclined to use the law of iterated expectations ($E[XY] = E[E[XY\mid A]]$)... since if you are given a value for $A$, then both $X$ and $Y$ become functions of $B$ and $C$ and therefore independent. Is this approach sound? Any other tips on how to calculate this Expectation? Thank you!
Hint: $XY$ is a product of $X$ and $Y$. You already have the definition of $X$ and $Y$ through i.i.d variables $A, B$ and $C$. Substitute, use linearity and remember, what is the definition of variance and how independent condition influences the calculation of expectation