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Given that $X\ge 0, q>0 $ and $P$ is the probability measure, I need to prove the following in a probability theory perspective:

  1. $E X = \int_{0}^\infty P(X>x) \,dx$
  2. $EX = \int_{0}^\infty xf(x) \,dx$
  3. $EX^q = \int_{0}^\infty qx^{q-1}P(X>x) \,dx$

In proving 1, I want to start with the definition of expected value which is: $EX = \int_{\Omega} X \,dP$, but not sure how to proceed. I checked with the other examples on this site describing this, but they use the existence of density function to prove 1. Can we prove 1 without using it? Also I think if I can prove 3 first, I can get 1 by setting $q=1$. Please help

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    Someone down-voted this. Could they explain why?2012-11-28
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    OK, my answer has now been updated to include a sketch of a proof of 2. It also shows how 2 entails 1, and how 1 entails 3.2012-11-30

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