X and Y have a bivariate normal distribution. I am given that $E[X] = 4$ and $E[Y] = 10$. I am asked to find $E[X^2 - Y^2]$ WITHOUT integration. I know how to solve for this using integration, but how can I find the solution without doing so?
2026-05-06 07:58:35.1778054315
Finding the expectation of functions of random variables with a bivariate normal distribution
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$$E[X^2-Y^2] = E[X^2] - E[Y^2] = (\operatorname{Var}(X)+E[X]^2)-(\operatorname{Var}(Y)+E[Y]^2) $$ $$ = 9+16-[61+100]$$
The first equality follows from the fact that the expectation of a sum is the sum of an expectation. The second follows from the "computational formula" for variance: $\operatorname{Var}(X) = E(X^2) - [E(X)]^2$.