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I have come up with 2 approaches but I'm not sure which one would be more appropriate.

1) $E[XY] = \sum_{m = 1}^{\infty} kP(X = k) * \sum_{n = 1}^{\infty} tP(Y=t)$

2) $E[XY] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}xyf_{X,Y}(x,y)dxdy$

I am asked to prove the following.

$$E[XY] = \sum_{m=1}^{\infty}\sum_{n=1}^{\infty} P(X \geq m, Y \geq n)$$

statsguy21
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    Obviously, if you are asked to prove this, then X and Y must be discrete, and even, nonnegative and integer valued, hence (2) is irrelevant while (1) (once suitably rewritten, the current version being absurd) is one way to go (not the best, but...). – Did Feb 08 '19 at 14:31
  • What approach would you suggest to go about proving the statement? Also, why would you say that (1) has to be rewritten? – statsguy21 Feb 08 '19 at 14:55
  • (1) is unsound because the indices in the sums do not fit ($m$ and $n$ vs. $k$ and $t$). – Did Feb 08 '19 at 15:01

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