A local pharmacy has two checkout stations available to its customers: a regular checkout station and an express checkout station. Customers with six or fewer items are assumed to join the express line. Let X and Y be the numbers of customers in the regular checkout line and the express checkout line, respectively, at the busiest time of a typical day. Note that these numbers include the customer(s) being served, if any. The joint distribution for X and Y is given in the file S04_30.xlsx.
a. Find the marginal distributions of X and Y. What does each of these distributions tell you?
b. Find the conditional distribution of X given Y. What is the practical benefit of knowing this conditional distribution?
c. What is the probability that no one is waiting or being served in the regular checkout line?
d. What is the probability that no one is waiting or being served in the express checkout line?
e. What is the probability that no more than two customers are waiting in both lines combined?
f. On average, how many customers would you expect to see in each of these two lines during the busiest time of day at the pharmacy?