Suppose I have a series of random variable $(X_1,...,X_k)$. They are independent but not identically distributed. I want to calculate the expected value of $X_1$ given the specific order of random variables. For instance, $X_1>...>X_k$.
I wonder if $E(X_1|X_1>\max_{i\ne 1} X_i) = E(X_1|X_1>X_2>...>X_k)$. I am asking this because when $k$ is large (more than 30) I cannot get the approximation by simulation with reasonable numbers.
While intuitively one might say that they should be equal since the extra information about the ordering of the random variables ($X_2, ..., X_n$) is independent of $X_1$, that is not really so. As a simple counterexample, consider $X_1$ is uniform on $\{0, 3\}$, $X_2$ is uniform on $\{2, -2\}$, and $X_3$ is uniform on $\{-1, 1\}$. Now, there is a case in which $X_1 > \max_{i \neq 1} X_i$, but $X_1 = 0$, namely (0, -2, -1). However, if you condition on $X_1 > X_2 > X_3$, then, $X_1 = 3$ a.s.
If you made them i.i.d., the proposed equality would have been true by symmetry.
