Chapter 6.3, Problem 10E

The total time from arrival to completion of service at a fast-food outlet, Y₁, and the time spent waiting in line before arriving at the service window, Y₂, were given in Exercise 5.15 with joint density function

$f(y_1,y_2)=\{\begin{array}{ll}{e}^{-y_1},\hfill & 0\le y_2\le y_1<\infty ,\hfill \\ 0,\hfill & \text{elsewhere}.\hfill \end{array}$

Another random variable of interest is U = Y₁ − Y₂, the time spent at the service window. Find

1. a the probability density function for U.
2. b E(U) and V(U). Compare your answers with the results of Exercise 5.108.