# Compute Expectation of Product Using Joint Survival Function

I know that, for a nonnegative random variable $$X$$, $$E[X] = \int x dF(x) = \int S(x) dx$$ where $$F(x)$$ and $$S(x)$$ are the CDF and survival function of $$X$$, respectively.

This was derived using integration by parts, which I only know for one dimension. Is there a corresponding result in the two dimensional case?

For nonnegative random variables $$X$$ and $$Y$$, is $$E[XY] = \int \int xy dH(x,y)$$ equivalent to something like $$E[XY] = \int \int S(x,y) dxdy .$$ where $$H(x,y)$$ and $$S(x,y)$$ are the joint CDFs and survival functions of $$X$$ and $$Y$$?

• You might be interested in stats.stackexchange.com/q/346856/119261. Commented Sep 5, 2022 at 15:44
• Because (assuming $X$ and $Y$ have finite variances) $E[XY] = E[(X+Y)^2/4] - E[(X-Y)^2/4],$ this is really just two univariate integrals.
– whuber
Commented Sep 5, 2022 at 16:12
• Doesn't that mean you have to write $S_{(X+Y)^2/4}$ in terms of $S_{X,Y}$? Commented Sep 5, 2022 at 18:10
• Yes: apply any of your favorite formulas.
– whuber
Commented Sep 6, 2022 at 15:12