Reference for expectation and variance of the product of independent random variables

Question

Given two independent random variables X, Y, the expectation of their product XY is:

$\mathrm{E}[XY] = \mathrm{E}[X]\cdot\mathrm{E}[Y]$

Similarly, the variance of the product of these variables is:

$\mathrm{Var}[XY] = \mathrm{Var}[X]\cdot \mathrm{Var}[Y] + \mathrm{Var}[Y]( \mathrm{E}[X])^2 + \mathrm{Var}[X] (\mathrm{E}[Y])^2$

While proofs or sketches of proofs can be found online (even within this forum), I have been struggling to find a citable reference of the above formulas (i.e., a textbook or a paper). Can you provide a suitable reference?

Answer 1

The first formula for independent random variables $X$ and $Y$, sometimes called the product law of expectation, is not hard to find. It can be found in most undergrad probability and statistics textbooks. For instance, you can find it here on page 160 of Introduction to the Theory of Statistics by Mood-Graybill-Boes, 3rd edition.

The second formula, even if not included in the main material of textbooks, could be found as an exercise, like this one here on page 89, problem I-32 of Exercises in Probability by T. Cacoullos.

Answer 2

On the Exact Variance of Products by Leo A. Goodman, published in Journal of the American Statistical Association, Volume 55, 1960 - Issue 292

is a peer-reviewed article focusing on this problem with the result for two independent variables stated in its Equation 2.