# The time between process problems

The time between process problems in a manufacturing line is exponentially distributed with a mean of 30 days.

(a) Let T be the waiting time (in days) for four problems. What is the distribution of T?

(b) What is the expected waiting time for four problems?

(c) What is the probability that the time until the fourth problem exceeds 120 days?

(d) Suppose that the problems can be classified into three mutually exclusive classes: I, II and III. The probability that a problem is of type I, II, and III, are respectively, 0.75, 0.2, 0.05.

(i) Give the distribution of the waiting time for a process problem of type I.