X is a random variables not negative satisfy $E[X^a]<\infty$ with $a>0$.Prove $$E[X^a]=a\int\limits_{0}^{\infty} x^{a-1}(1-F(x))dx$$.
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Start with the definition of $E[X^a]$ and try integration by parts ... Please post your answer. – Semoi Mar 21 '20 at 17:03
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Note that $$ X^a=\int_{0}^\infty a x^{a-1 } I(X>x)\, dx$$ Take expectations of both sides and use Tonell's theorem to deduce the result.
Sri-Amirthan Theivendran
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