# Computing the n-dimensional integral of of guassian curve

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.

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.