Prove that for any random variable Z taking values in the range {1, 2, 3, . . .}, E(Z) = (Σ from i = 1 to infinity) P(Z ≥ i)?

Asked Mar 02 '17 at 20:32

Active Mar 02 '17 at 20:32

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How would I even begin proving this? I've been thinking about this for an hour now with no luck.

I attempted to get E(Z) and go from there, but I can't seem to prove it

asked Mar 02 '17 at 20:32

Try rearranging each sum and see both sums have the same terms. – Jonathaniui Mar 02 '17 at 20:40
Hint: $P(i)=P(Z=i)=P(Z≥i)-P(Z≥i+1)$. Thus, say, $1\times P(1)+2\times P(2)=P(Z≥1)-P(Z≥2) + 2P(Z≥2)-2P(Z≥3)=P(Z≥1)+P(Z≥2)-2P(Z≥3)$ and so on. – lulu Mar 02 '17 at 20:43

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