 Solution

For $0\le u\le 1$, independence givesThus $U$ has probability density function $f_U(u)=2(1-u)$, and its expected value isPointwise, $U+V=X+Y$ and $UV=XY$. By linearity of expectation, $\mathbb{E}V=1-1/3=2/3$; by independence, $\mathbb{E}(UV)=\mathbb{E}X\mathbb{E}Y=1/4$. Consequently the covariance of two uniform order statistics isThe order statistics are dependent despite the independence of the original observations.