Sunday, 9 June 2019

probability - Proof: Xge0,r>0RightarrowE(Xr)=rintinfty0xr1P(X>x)dx




As the title states, the problem at hand is proving the following:



X0,r>0E(Xr)=r0xr1P(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=rX0xr1dx=r+0xr11X>xdx.


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