# Conditional expectation of the maximum of two independent uniform random variables given one of them

Given $X_1$ and $X_2$ two independent random variables both uniformly distributed on $[0,1]$. What is the conditional expectation of $\max\{X_1,X_2\}$ given $X_2$? And the conditional expectation of $\min\{X_1,X_2\}$ given $X_2$?

• Given $X_2$, what is the probability that $X_1$ is smaller? And what is the expected value of $\max(X_1,X_2)$ in that case? And what if the opposite is true? – Harald Hanche-Olsen Oct 5 '14 at 22:26

$$\Pr(\max\{X_1,X_2\} = X_2\mid X_2=x)=\Pr(X_2>X_1\mid X_2=x) = \Pr(x>X_1)=x.$$ The probability distribution of $\max$ given that $\max\ne X_2$ and $X_2=x$ is the probability distribution of $X_1$ given $X_1>x$, so that is uniform on $[x,1]$.
Therefore $$\mathbb E(\max\mid X_2) = \dfrac{1+X_2^2}2.$$