Covariance of max and min of two uniformly distributed random variables

Question

Let X, Y be two independent random variables following a uniform distribution in the interval (0,1). Let U=Min(X,Y), and V=Max(X,Y). How do I find the Covariance(V,U)? Since X and Y are i.i.d., I guess $F_U(u)= 2F_x(u)-F_x^2(u)$, thus E(U)=1/3; on the other hand, I should have $F_V(v)=F_x^2(v)$, with E(V)=2/3. How do I find the joint distribution $F_{UV}(u,v)$ and thus E(UV)?

Answer 1

You need not to find joint distribution. Note that $UV=XY$, so $$\mathbb E(UV)=\mathbb E(XY)=\mathbb E(X)\mathbb E(Y)=\frac14.$$