1. Suppose that four calls per minute arrive at a switchboard and that the arrival of calls follows a Poisson process. What is the probability that the next call will arrive within seven minutes? Suppose that a malfunction dis- ables the switchboard for two minutes. What is the probability that no calls arrive during this period?

2. System failures at an industrial facility are approximately Poisson dis- tributed with λ = 0.25 per hour (the system experiences a failure every four hours). If system startup begins at 7 A.M. on a typical work day and T is the time until the first failure occurs, find

(a) The probability that it is at least two hours until the first failure occurs

(b) The probability that it is no more than six hours until the first failure occurs

(c) The average time to the first failure

(d) The probability that the time to the next failure exceeds the average time to failure.