How does one calculate the following:
Given $X_1,...,X_n$ are i.i.d. uniform random variables, calculate $E[X_1|X_{(n)}]$ where $X_{(n)}$ denotes the $n^{th}$ order statistic; and
Given $X_1, ..., X_n$ are i.i.d. random variables, calculate $E[X_1 | \bar{X}_n]$.
First question is answered here: Conditional expectation to de maximum $E(X_1\mid X_{(n)})$
As to the second question, note that $$ \frac{1}{n}\sum_{k=1}^nE(X_k|\bar{X}_n) = E\left(\frac{1}{n}\sum_{k=1}^n X_k\big| \bar{X}_n \right) = E(\bar{X}_n|\bar{X}_n)= \bar{X}_n. $$ Then realize that $E(X_1|\bar{X}_n)=\cdots =E(X_k|\bar{X}_n)$, implying that $$ E(X_1|\bar{X}_n) =\bar{X}_n $$
