# real analysis – Solving \$int_0^infty N(1-F(t))^{N-1}tf(t)dt\$ when the expected value is known

Suppose that $$f:mathbb R_{geq 0} rightarrow mathbb R_{geq 0}$$ is a probability density function, and $$F$$ is a cumulative distribution function (i.e. $$F(t)=int_0^t kf(k)dk$$). Also, assume that the expected value, $$E = int_0^infty tf(t)dt$$ is known.

(Question 1) Is there a way to solve $$A_N:=int_0^infty N(1-F(t))^{N-1}tf(t)dt$$, where $$N$$ is a positive integer, so that the result only depends on $$N$$ and $$E$$ (e.g. $$Ee^{-N}+const$$)? If so, how?

(Question 2) Or if solving exactly is hard or impossible, then would it possible to get an approximate solution?

Also, from the problem I am working on, it is safe to assume that $$A_Nrightarrow C$$ as $$Nrightarrow infty$$ where $$0 is some constant, and $$A_{N}$$ is decreasing. Not sure whether these are helpful, though…

So far, I tried to approximate $$(1-F(t))^{N-1}$$ as $$e^{-F(T)(N-1)}$$, but haven’t gotten much so far…