One of the earliest applications of the Poisson distribution was in analyzing incoming calls to a telephone switch-board. Analysts generally believe that random phone calls are Poisson distributed. Suppose phone calls to a switch-board arrive at an average rate of 2.4 calls per minute.
a. If an operator wants to take a one-minute break, what is the probability that there will be no calls during a one-minute interval?
b. If an operator can handle at most five calls per minute, what is the probability that the operator will be unable to handle the calls in any one-minute period?
c. What is the probability that exactly three calls will arrive in a two-minute interval?
d. What is the probability that one or fewer calls will arrive in a 15-second interval?