As the title states, the problem at hand is proving the following:
X≥0,r>0⇒E(Xr)=r∫∞0xr−1P(X>x)dx
Attempt/thoughts on a solution
I am guessing this is an application of Fubini's Theorem, but wouldn't that require writing P(X>x) as an expectation? If so, how is this accomplished?
Thoughts and help are appreciated.
Answer
Proof: Consider the expectation of the identity
Xr=r∫X0xr−1dx=r∫+∞0xr−11X>xdx.
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