The number of customers who use a drive-thru Automated Teller Machine (or ATM) varies from one hourto the next, with a rate of 7 customers every 15 minutes.
(a) A random variable X is to measure the number of customers who use this drive-thru ATM in a 15- minute interval. What is the probability that between 8 and 10 customers, inclusive, use this ATM in a given 15-minute interval?
(b) What is the probability that the duration of time passing between successive customers using this ATM is between 2 and 5 minutes?
(c) A customer has just driven away from this ATM, and there are no vehicles in the drive-thru ATM line. If at least 2 minutes pass until the next customer enters the drive-thru ATM line, what is the probability that in total, no more than four minutes pass until the next customer enters the drive-thru ATM line?
(d) Over the course of a 5-day week, forty-five non-overlapping 15-minute intervals were randomly chosen and the number of customers using the drive-thru ATM is to be observed in each of these 15-minute intervals. What is the probability that average number of customers using the drive-thru ATM in these forty-five 15-minute intervals will be between 6.5 and 7.75 customers?