Expectation of a Product of Independent Random Variables

Let $X$ and $Y$ be independent random variables on a probability space $(\Omega,\mathcal{F},P)$, each with finite expectation. Then the product $XY$ (a random variable, since $XY=\tfrac{1}{4}\bigl((X+Y)^2-(X-Y)^2\bigr)$ and sums, differences, and squares of random variables are random variables by Step 0(a) of the proof of Linearity and Monotonicity of the Lebesgue Integral and the power argument of Expectation, Variance, and Moments) has finite expectation, and

Consequently, if $X_1,\dots,X_r$ are independent random variables each having finite expectation and finite second moment, then for $i\ne j$,

and the variance is additive over independent summands: