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Let $x_1,x_2,...,x_n$ be $n$ instantiations of random variable $X$ having the cdf $F(x)$ and pdf $f(x)$,taking values in the interval $ (a,b) $.Is the expresion correct : $$ P(x_1 \leq x_2 \leq x_3 \dots\leq x_n)=\int_{x_n=a}^b\quad \int_{x_{n-1}=a}^{x_n}\dots \int_{x_1=a}^{x_2} \prod_{i=0}^nf(x_i) dx_1 dx_2 \dots dx_n $$. How do we compute the above integral for some given $f(x)$.In particular if $X$ is normally distributed so that $$ f(x_i)=\frac{1}{\sqrt{2 \pi}} e^{-x_i^2/2},i=1,2,3,, \dots,n ,$$ how do we evaluate or find the asymptotic value of the above integral when,say, .Thanks for any help/hints/responces in advance.

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Assuming continuous random variables, this integral will evaluate to $1/n!$ regardless of distribution, because what you ask for is one of the n possible permutations of a random sample.

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    $\begingroup$ that is really elegant.Could you ,however,just give a hint or sketch about how to elavuate this integral numerically to corroborate the claim experimentally as well? $\endgroup$ Jul 4, 2021 at 19:35
  • $\begingroup$ Better, you can do simulations. Multi-dimensional integral estimation can be time consuming, but simply you can try grid estimation, e.g. with dx=dy=...=0.01 steps etc. $\endgroup$
    – gunes
    Jul 4, 2021 at 19:35

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