Prove the reproductive property of independent Poisson RVs. Hence find the probability of 5 or more telephone calls arriving in a 9 min period in a college switch-board, if the telephone calls that are received at the rate of 2 every 3 min follow a Poisson distribution.

A service station has a pump that distributes diesel fuel to automobiles. The station owner estimates that only about 3.2 cars use the diesel pump every 2 hours. Assume the arrivals of diesel pump users are Poisson distributed. a. What is the probability that three cars will arrive to use the diesel pump during a one-hour period?     b. Suppose the owner needs to shut down the diesel pump for half an hour to make repairs. However, the owner hates to lose any business. What is the probability that no cars will arrive to use the diesel pump during a half-hour period?     c. Suppose five cars arrive during a one-hour period to use the diesel pump. What is the probability of five or more cars arriving during a one-hour period to use the diesel pump?

4. The owner of a tree farm specializes in growing pine trees. They are grown in rows of 50 trees and the owner has found on the average, about 4 trees per row will not be suitable to sell. Assuming that the Poisson distribution is applicable here, find the probability that a row of tree selected for shipment will contain 2 unsaleable trees.

Assume that when a human resource managers are randomly selected 51% SAP applicant should follow up within two weeks and 14 human resource managers are randomly selected the probability that viewer than three of them say job applicants should follow up within two weeks

It is known that the frequency of a database server experiencing a Poisson distributed error with an average of 2 times a week.a. What is the probability that in two weeks there will be three errors?b. What is the probability that after one day a database error occurs, the next error will occur less than four days later?c. It is known that on the third day there is not a single error, what is the probability that on the fifth day there is still no error?