# 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 $$A_N\rightarrow C$$ as $$N\rightarrow \infty$$ 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...
• Obviously, the exact value can be different for the same fixed $N\ge 2$ and $E$ (just plug in two favorite random variables of yours with the same expectation), so apparently by Q1 you meant something different from what you wrote. What was it? – fedja Sep 8 at 18:34