I'm stuck on the following probability problem and would welcome any help:
Consider three random uniform variables on [0,1]. Let X be the minimum of these three variables and Y be the maximum. What is the expected value of X*Y?
Am I right in assuming I need to find the joint distribution function of X and Y? I have found the separate distribution functions but I'm having trouble with the joint one since X and Y are not independent.
It's simpler to deal with the three uniform random variables $U_1, U_2, U_3$. Given $U_1 < U_2 < U_3$, $XY = U_1 U_3$. So $$\eqalign{ \mathbb E [XY | U_1 < U_2 < U_3] &= \mathbb E[U_1 U_3 | U_1 < U_2 < U_3]\cr &= 6 \int_0^1 du_3 \int_0^{u_3} du_2 \int_0^{u_2} du_1 \; u_1 u_3}$$ By symmetry, the result is the same for all the other orderings of $U_1, U_2, U_3$, so this is also $\mathbb E[XY]$.