1. ## Conditional expectation

Let $\displaystyle \xi_1,\xi_2,\xi_3$ be three independent, identically distributed and bounded random variables with density $\displaystyle p(x)$.

What is the distribution of: $\displaystyle \mathbb{E}\big(\max(\xi_1,\xi_2,\xi_3)|\min(\xi_1, \xi_2,\xi_3)\big)$, as a function of $\displaystyle p$ ?

One can easily obtain the distribution of $\displaystyle \max(\xi_1,\xi_2,\xi_3)$ and $\displaystyle \min(\xi_1,\xi_2,\xi_3)$. But how do you get the joint distribution?
Unless there is a better way to proceed.

2. Let $\displaystyle M=\max(\xi_1,\xi_2,\xi_3)$ and $\displaystyle m=\min(\xi_1,\xi_2,\xi_3)$. You can start from there:

$\displaystyle P(t\leq m\mbox{ and }M\leq u)=P(t\leq \xi_1\leq u)^3$.

This gives the distribution of $\displaystyle (m,M)$.

3. you can use the 'multinomial' formula for joint densities of order stats

$\displaystyle f(x_1,x_3)=3!f(x_1)[F(x_3)-F(x_1)]f(x_3)$

where I'm using $\displaystyle x_1$ as the first order stat and $\displaystyle x_3$ as the third.

4. This is what you get using the joint probability suggested by Laurent.

I am not familiar with the multinomial formula you mention. Is there any reference you could suggest where I can find it?