# Given that X and Y are normally distributed as N(0,3) and N(0,5) respectively, what is the expected value of (XY)^2?

Given that X and Y are independent and normally distributed as N(0,3) and N(0,5) respectively, what is the expected value of (XY)^2?

• Are $X$ and $Y$ independent? Jan 24 at 11:52
• @Henry yes, they are Jan 24 at 12:16

• $$E[XY] = E[X]\cdot E[Y] \quad \text{if X and Y are independent}$$
• $$E[X^2] = \text{Var}[X]+ E[X]^2$$
• @Jacques That extra rule is not needed and applying it only complicates the solution. Another one that is used implicitly is that functions of independent variables (separately) remain independent. For this specific question I would point out that $E[X^2]=\operatorname{Var}(X)$ because $E[X]=0.$ That eliminates all probability calculations and allows one to write down the answer immediately, depending on how "N(0,3)" is interpreted (is its variance $3$ or $9$?). (+1)
• @JacquesWainer: The more useful rule is that $(XY)^2 = X^2Y^2$. Jan 24 at 19:53