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.

Thanks in advance for your help.