Expectation of Max and Min of Two Uniform Random Variables

Question

I'm looking for someone to walk me through how to solve E[ZW] if Z = max[X,Y] & W = min[X,Y]. X & Y are independent, uniform variables over the interval [0,1].

I solved E[Z] = 2/3 and E[W] = 1/3.

E[ZW] should equal 1/4 but I keep getting the wrong answer. I have searched the forums and used other people's answers to try and solve it but have gotten the incorrect numbers.

Thank you in advance!

Answer 1

Notice that $$\max(X,Y) \times \min(X,Y)= XY$$

Hence $$\mathbb{E}[ZW] = \mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y]$$

Answer 2

Here's a hint: $ZW = XY$ no matter whether $Z > W$, $Z < W$, or $Z = W$.