TITLE: Conditional expectation to de maximum $E(X_1\mid X_{(n)})$
QUESTION [10 upvotes]: Let $X_1, \ldots, X_n$ a random sample of a Uniform(0,1):
Which is $E(X_1\mid X_{(n)})$ ?
where $X_{(n)}=\max\{X_1,\ldots,X_n\}$

REPLY [4 votes]: You're looking for $E(X_1\mid \max)$.  The probability that $X_1=\max$ is $1/n$. So
$$
E(X_1\mid \max) = \frac 1 n E(X_1\mid X_1=\max, \max) + \frac{n-1} n E(X_1\mid X_1\ne \max, \max).
$$
The first term is easy to find: $E(X_1\mid X_1=\max,\max) = \max$, so we have
$$
E(X_1\mid \max) = \frac 1 n \max + \frac{n-1} n E(X_1\mid X_1\ne \max, \max).
$$
Then you need to show that the conditional distribution of $X_1$ given $\max$ and given the event that $X_1\ne\max$ is uniform on the interval from $0$ to $\max$.