Suppose we have $n$ normal variable $X_1,X_2,\dots,X_n$, with corresponding mean $\mu_1,\dots,\mu_n$ and sd $\sigma_1,\dots,\sigma_n$. What is the probability of $X_1 < X_2 < \dots < X_n$, i.e. $P(X_1 < X_2 < \dots < X_n)$.
Numerical method is also okay. Actually I have thought about how to do the multi-variable integration but I cannot get it done. When $n=2$, it is easy to see that $X_1-X_2$ is also a normal variable so $P(X_1-X_2<0)$ is quite easy to calculate. However, I think when $n=3$, because of $P(X_1-X_2)$ and $P(X_2-X_3)$ are not independent, the result is not easy to calculate. Currently I am thinking about whether we can use some method to approximately calculate the value the that probability. I need to come up with an algorithm to compute that.