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Would you give me any suggestions or comments on evaluating the following $n$-dimensional integral? $$ \int_{[0,t]^n} h(x) dx $$

where $ x=(x_1 ,x_2 , \cdots, x_n ), h(x)= \prod_{k=1}^n min( \bar{x}_k), \bar{x}_k =\{x_1,x_2, \cdots,x_n \} - \{x_k \}$

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  • $\begingroup$ where does this integral arise? $\endgroup$
    – Per Alexandersson
    Commented Oct 8, 2014 at 7:07
  • $\begingroup$ If $h(x)=min \{x_1,x_2,\cdots, x_n \}$, I can easily handle it. But I wonder that there is(are) any nice ways to get the value of the problem given above. $\endgroup$
    – hkju
    Commented Oct 8, 2014 at 7:08
  • $\begingroup$ I don't get it anywhere from. I just consider it for fun. $\endgroup$
    – hkju
    Commented Oct 8, 2014 at 7:13
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    $\begingroup$ The integrand is symmetric, so the value of the integral on $I^n$ is $n!$ times the value of the integral of $x_2 x_1^{n-1}$ on the simplex $\{0<x_1<x_2<\dots<x_n<1\}$, that you can easily compute by iterate integration. Should be $\frac{1}{2n\choose n+1}$. $\endgroup$
    – Pietro Majer
    Commented Oct 8, 2014 at 7:31
  • $\begingroup$ Thanks, Pietro. How about the case $h(x)=min(x_1,x_2) min(x_2,x_3) \cdots min(x_{n-1},x_n) min(x_n,x_1)$ ? $\endgroup$
    – hkju
    Commented Oct 8, 2014 at 7:52

1 Answer 1

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This is $\mathbb E (X_1^{n-1} X_2) $ where $ X_i $ is the $ i $ th order statistic of a sample from the uniform distribution on $[0, t] $. To evaluate it can try using the joint pdf of these order statistics, given at http://en.m.wikipedia.org/wiki/Order_statistic

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