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Is there any general closed form to the following? $$\int^1_0 f^2(x)F^{n-1}(x)dx$$ when F is a CDF with support $[0,1]$ and $f$ is its corresponding pdf.

It's easy when it is $\int^1_0 f(x)F^{n-1}(x)dx$, but what happens when the pdf is powered by some number?

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  • $\begingroup$ The integral $\int_0^1 f(x)F^n(x)dx$ has a nice probabilistic interpretation; iff $X_0,X_1,\dots,X_n$ are iid with pdf $f$, then that integral computes the probability that $X_0$ is the smallest value (think of the integral as conditioning on the value of $X_0=x$, and $F^n(x)$ is the probability all of $X_1,\dots,X_n$ are smaller). However, I cannot imagine a situation where $f^2(x)$ would have any meaning, so it is unlikely there is a similar general nice answer. $\endgroup$ Commented Jun 28, 2018 at 19:59

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