   Chapter 5.1, Problem 5E

Chapter
Section
Textbook Problem

The number of customers waiting for gift-wrap service at a department store is an rv X with possible values 0, 1, 2,3, 4 and corresponding probabilities .1, .2, .3, .25, .15. A randomly selected customer will have 1, 2, or 3 packages for wrapping with probabilities .6, .3, and .1, respectively. Let Y = the total number of packages to be wrapped for the customers waiting in line (assume that the number of packages submitted by one customer is independent of the number submitted by any other customer).a. Determine P(X = 3, Y = 3), i.e., p(3, 3).b. Determine p(4, 11).

a.

To determine

Find P(X=3,Y=3), that is, p(3,3).

Explanation

Given info:

The random variables are X and Y. X is the number of customers waiting for gift-wrap service, with possible values 0, 1, 2, 3 and 4 with corresponding probabilities 0.1, 0.2, 0.3, 0.25 and 0.15. Y is the total number of packages to be wrapped for the customers waiting in line, where each customer may have 1, 2 or 3 packages to be wrapped with respective probabilities 0.6, 0.3 and 0.1. The number of packages for one customer is independent of that of the other.

Calculation:

The probability P(X=3,Y=3) or p(3,3) is the probability that “there are X=3 customers waiting for the gift-wrap service” and “a total of Y=3 packages are to be wrapped for the customers waiting in line”.

It is known that each waiting customer has either 1 or 2 or 3 packages to be wrapped.

Thus, there will be 3 customers waiting in line and a total of 3 packages waiting to be wrapped, if each of the three waiting customers has exactly 1 package to be wrapped.

Thus,

P(X=3,Y=3)=p(3,3)=P(3 customers having 1 package each)=P(

b.

To determine

Find p(4,11).

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