I have difficulties with an old exam problem :
Let X be a positive random variable defined on a probability space (Ω,F,P). Show that
∫∞0tkP(X≥t)dt=∫∞0∫Ωtk∫{X(ω)≥t}dtdP(ω) Infer from this the integral expression of E(Xk+1) (where E is the expectation)
We have Fubini theorem, which we can apply to a B(R)⊗F-measurable function because the Lebesgue measure is σ-finite and P is also σ-finite because it is a probability. I think we can write P(X≥t) as ∫{X(w)≥t}dP(ω) but I don't know how to proceed next. Especially I don't see how to introduce the ∫Ω.
Edit
From the comments, there must be an error in the description of the exam problem. It should have been the following :
Let X be a positive random variable defined on a probability space (Ω,F,P). Show that
∫∞0tkP(X≥t)dt=∫∞0∫Ωtk1{X(ω)≥t}dtdP(ω) Where 1{X(ω)≥t} is the characteristic function of {X(ω)≥t}Infer from this the integral expression of E(Xk+1) (where E is the expectation)
Answer
By Fubini's theorem, we have
∫∞0tkP(X⩾t)dt=∫∞0tkE[1{ω:X(ω)⩾t}]dt=∫∞0tk∫Ω1{ω:X(ω)⩾t}dPdt=∫Ω∫X(ω)0tkdtdP=∫Ω1k+1Xk+1(ω)dP(ω)=1k+1E[Xk+1].
Hence E[Xk+1]=(k+1)∫∞0tkP(X⩾t)dt.
The crucial part here is that 1{ω:X(ω)⩾t}(ω)=1{t:t⩽X(ω)}(t).
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