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.
(ii) Suppose that the problems I, II and III, cost (per problem): 1,000, 2,500 and 6,000 $, respectively. Give the mean total cost for process problems for a period of 90 days.
(e) There are no process problems for 30 days, what is the probability that there will be at least one process problem in the next 30 days?
For (a) (b) and (c) I have managed to answer the following:
(a) The waiting time distribution for T is Erlang since this is a sum of exponential distributions.
(b) E[X] = r.1/lambda = 4/lambda , where lambda = 1/30 ==> 4/0.033
(c) This would be a Poisson distribution P(X>120)
However for (d) and (e) I am kind of stuck. Any help would be greatly appreciated!