Variance of a product of independent variables

Question

Independent random variables $X$ and $Y$ satisfy $\mathbb{E}X=4$ , $\mathbb{E}X^2=20$ , $\mathbb{E}Y=-3$ and $\mathbb{E}Y^2=45$ . What is $\operatorname{Var}(XY)$ ?

Short answer

$756$ . Independence lets moments factor: $\mathbb{E}[X^2Y^2]=20\cdot45=900$ and $\mathbb{E}[XY]=-12$ . So the variance is $900-144=756$ .

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Lessons behind this question

Maximum Likelihood Estimation: Score and Log-Likelihood
Independence
Covariance and Correlation

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