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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 $Y$?

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    $\begingroup$ You might be interested in stats.stackexchange.com/q/346856/119261. $\endgroup$
    – StubbornAtom
    Sep 5, 2022 at 15:44
  • $\begingroup$ 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. $\endgroup$
    – whuber ♦
    Sep 5, 2022 at 16:12
  • $\begingroup$ Doesn't that mean you have to write $S_{(X+Y)^2/4}$ in terms of $S_{X,Y}$? $\endgroup$
    – Peter_Pan
    Sep 5, 2022 at 18:10
  • $\begingroup$ Yes: apply any of your favorite formulas. $\endgroup$
    – whuber ♦
    Sep 6, 2022 at 15:12

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