Chapter 5.8, Problem 108E

If Y₁ is the total time between a customer’s arrival in the store and departure from the service window and if Y₂ is the time spent in line before reaching the window, the joint density of these variables was given in Exercise 5.15 to be

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

The random variable Y₁ − Y₂ represents the time spent at the service window. Find E(Y₁ − Y₂) and V(Y₁ − Y₂). Is it highly likely that a randomly selected customer would spend more than 4 minutes at the service window?

5.15 The management at a fast-food outlet is interested in the joint behavior of the random variables Y₁, defined as the total time between a customer’s arrival at the store and departure from the service window, and Y₂, the time a customer waits in line before reaching the service window. Because Y₁ includes the time a customer waits in line, we must have Y₁ ≥ Y₂. The relative frequency distribution of observed values of Y₁ and Y₂ can be modeled by the probability density function

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

with time measured in minutes. Find

1. a P(Y₁ < 2, Y₂ > 1).
2. b P(Y₁ ≥ 2Y₂).
3. c P(Y₁ − Y₂ ≥ 1). (Notice that Y₁ − Y₂ denotes the time spent at the service window.)