Chapter 5.3, Problem 33E

Suppose that Y₁ is the total time between a customer’s arrival in the store and departure from the service window, Y₂ is the time spent in line before reaching the window, and the joint density of these variables (as was given in Exercise 5.15) is

$f\left(y_1,y_2\right)=\{\begin{array}{l}{e}^{-y_1},0\le y_2\le y_1\le \infty ,\\ 0,\text{elsewhere}\text{.}\end{array}$

1. a Find the marginal density functions for Y₁ and Y₂.
2. b What is the conditional density function of Y₁ given that Y₂ = y₂? Be sure to specify the values of y₂ for which this conditional density is defined.
3. c What is the conditional density function of Y₂ given that Y₁ = y₁? Be sure to specify the values of y₁ for which this conditional density is defined.
4. d Is the conditional density function f(y₁|y₂) that you obtained in part (b) the same as the marginal density function f₁(y₁) found in part (a)?
5. e What does your answer to part (d) imply about marginal and conditional probabilities that Y₁ falls in any interval?

5.15 The management at a fast-food outlet is interested in the joint behavior of the random variables Y₁, defined as the total lime between a customer’s arrival at the store and departure from the service window, and Y₂, the lime 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

$f\left(y_1,y_2\right)=\{\begin{array}{l}{e}^{-y_1},0\le y_2\le y_1\le \infty ,\\ 0,\text{elsewhere}\text{.}\end{array}$

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.)