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I have to show the following inequality: $$ \sum_{k=0}^{\infty} P[Y>k] \leq E[Y] \leq \sum_{k=1}^{\infty} P[Y>k]$$ under the condition that Y is a non negative continuous random Variable.

My first idea was to rewrite the inequality using integrals, but then i end up having to integrate over a sum, which is nothing im particulary excited about. Any Ideas on how i can do this?

Dominik
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Killercat
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  • This has been asked many times. See both the continuous and discrete cases in http://math.stackexchange.com/questions/1795529/why-is-mathbbex-1-sum-infty-k-1-mathbbpx-k-true –  Dec 07 '16 at 12:39
  • Also: http://math.stackexchange.com/q/64186/9464 –  Dec 07 '16 at 12:43

1 Answers1

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Hint: If $Y$ is nonnegative, then $$E[Y] = \int Y \, dP = \int \int_0^\infty I\{x < Y\} \, dx \, dP = \int_0^\infty \int I\{x < Y\} \, dP \, dx = \int_0^\infty P(Y > x) \, dx$$

Now apply the standard comparison between integrals and sums.

Dominik
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  • are you - by any chance - talking about this one: $ \sum_{i \in I} f(i) = \int_I {f(x) #dx} $ – Killercat Dec 07 '16 at 12:36
  • No, that is not a valid equation. I am talking about a comparison between Riemann-integral and infinite sum. – Dominik Dec 07 '16 at 12:41