# Are $E(X_1|X_1>\max_{i\ne1}X_i)$ and $E(X_1|X_1>X_2>...>X_k)$ the same?

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.

• They are not equal because the first does not assume an order on the rest of the variables. Commented May 23, 2021 at 17:18

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.