Calculate the conditional expectation of X given the order statistics?

Question

Here is the problem:

Let $X_1,...,X_n$ be i.i.d. random variables and let $X_{(1)},...,X_{(n)}$ be the order variables. Determine: $ E[X\mid X_{(1)},...,X_{(n)}]. $

I know the conditional expectation formula, my problem is: how can I determine the pdf $f_{X,X_{(1)},...,X_{(n)}}$ or $f_{X\mid X_{(1)},...,X_{(n)}}$? If $X$ were independent from the order variables it would be easier, but I think that's not the case.

Answer 1

I think something along these lines would work:

Since $X_1,\ldots,X_n$ are i.i.d,

$$E\left[X_1\mid X_{(1)},\ldots,X_{(n)}\right]=E\left[X_2\mid X_{(1)},\ldots,X_{(n)}\right]=\cdots=E\left[X_n\mid X_{(1)},\ldots,X_{(n)}\right]$$

So,

$$E\left[\sum_{i=1}^n X_i\mid X_{(1)},\ldots,X_{(n)}\right]=\sum_{i=1}^n E\left[X_i\mid X_{(1)},\ldots,X_{(n)}\right]=nE\left[X_1\mid X_{(1)},\ldots,X_{(n)}\right]$$

In other words,

\begin{align} nE\left[X_1\mid X_{(1)},\ldots,X_{(n)}\right]&=E\left[\sum_{i=1}^n X_{(i)}\mid X_{(1)},\ldots,X_{(n)}\right] \\&=\sum_{i=1}^n E\left[X_{(i)}\mid X_{(1)},\ldots,X_{(n)}\right] \\&=\sum_{i=1}^n X_{(i)} \end{align}

Hence,

$$E\left[X_1\mid X_{(1)},\ldots,X_{(n)}\right]=\frac1n\sum_{i=1}^n X_{(i)}=\frac1n\sum_{i=1}^n X_i$$