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

Question

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

Quizplus · Accepted Answer

The Answer is

E [ h ( X , Y ) ] = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } | x - y | \cdot 6 x ^ { 2 } y d x d y = 2 \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } ( x - y ) \cdot 6 x ^ { 2 } y d y d x

=

12 \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } \left( x ^ { 3 } y - x ^ { 2 } y ^ { 2 } \right) d y d x = 12 \int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { 6 } d x = \frac { 1 } { 3 }

Abby and Bianca Have Agreed to Meet for Lunch Between fx(x)={3x20≤x≤10 otherwise f _ { x } ( x ) = \left\{ \begin{array} { l l } 3 x ^ { 2 } & 0 \leq x \leq 1 \\0 & \text { otherwise }\end{array} \right.fx​(x)={3x20​0≤x≤1 otherwise ​

Abby and Bianca Have Agreed to Meet for Lunch Between $f _ { x } ( x ) = \left\{ \begin{array} { l l } 3 x ^ { 2 } & 0 \leq x \leq 1 \\0 & \text { otherwise }\end{array} \right.$