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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 $0<C<\infty$ 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...
Thank you in advance :)

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    $\begingroup$ 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? $\endgroup$ – fedja Sep 8 at 18:34

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