Recall in Exercise 5.32 that the number of customer arrivals at a bank’s drive- up window in a 15- minute period is Poisson distributed with a mean of seven customer arrivals per 15- minute period. Define the random variable x to be the time (in minutes) between successive customer arrivals at the bank’s drive- up window.

a. Write the formula for the exponential probability curve of x.

b. Sketch the probability curve of x.

c. Find the probability that the time between arrivals is:

(1) Between one and two minutes.

(2) Less than one minute.

(3) More than three minutes.

(4) Between 1 2 and 3 1 2 minutes.

d. Calculate m x, s x 2 , and s x . e Find the probability that the time between arrivals falls within one standard deviation of the mean; within two standard deviations of the mean.

In Exercise 5.32

A bank manager wishes to provide prompt service for customers at the bank’s drive- up window. The bank currently can serve up to 10 customers per 15- minute period without significant delay. The average arrival rate is 7 customers per 15- minute period. Let x denote the number of customers arriving per 15- minute period. Assuming x has a Poisson distribution:

a. Find the probability that 10 customers will arrive in a particular 15- minute period.

b. Find the probability that 10 or fewer customers will arrive in a particular 15- minute period.

c. Find the probability that there will be a significant delay at the drive- up window. That is, find the probability that more than 10 customers will arrive during a particular 15- minute period.