Question27. Annie and Alvie have agreed to meet for lunch between noon (0:00 P.M.) and 1:00 P.M. Denote Annie's arrival time by $X$. Alvie's by $Y$, and suppose $X$ and $Y$ are independent with pdf's $$\begin{array}{l} f_{X}(x)=\left\{\begin{array}{cl} 3 x^{2} & 0 \leq x \leq 1 \\ 0 & \text { otherwise } \end{array}\right. \\ f_{X}(y)=\left\{\begin{array}{rl} 2 y & 0 \leq y \leq 1 \\ 0 & \text { otherwise } \end{array}\right. \end{array}$$
What is the expected amount of time that the one who arrives first must wait for the other person? [Hint: $\quad h(X, Y)=|X-Y|$.