Bivariate Normal Expectation of Product of Squares Math, CS, Data
Assume $X$ and $Y$ are each standard normal, and they have correlation coefficient $\rho$. Find the expectation of $X^2Y^2$.
First, consider the properties of $Y$ for a fixed $X$. In such a case, we have a simple linear model, and we know that $\E [Y \vert X] = \rho X$. Furthermore, $\rho^2$ represents the proportion of variation in $Y$ that is explained by $X$, so the proportion unexplained is
We can easily extend this result to the more general case of nonstandard normality. Assume $A \sim N(\mu_A, \sigma_A^2)$ and $B \sim N(\mu_B, \sigma_B^2)$ and they have correlation coefficient $\rho$. Then we can equivalently reexpress them as $A = \mu_A + \sigma_A X$ and $B = \mu_B + \sigma_B Y$ for some standard normal $X$ and $Y$ with correlation $\rho$.
We will also need to find a couple more expectations along the way.