A representative of a high-speed Internet provider calls customers to assess their satisfaction with the service. It takes her 6 seconds to turn on a phone and dial a number; then 3 additional seconds to detect a busy signal, or 25 additional seconds to wait for 5 rings and conclude that no one will answer; and 1 second to end a call. After an unsuccessful call, she redials (in the course of several days) until the customer answers or she has dialed four times. The outcome of each dialing is determined in an identical way: the customer being called is using the line with probability 0. 2; or is unavailable to answer the call with probability 0.3; or is available and can answer the call within seconds, which is a continuous random variable with the mean of 12 seconds and the exponential distribution. (Note: it is possible for the customer to be available, but they take too long to answer the phone.) The calling process ends when the customer answers the call, or when four unsuccessful calls have been completed. Let denote the total time spent by the representative on calling one customer. Your objective is to estimate several statistics of . Toward this end, perform the following.

Design a Monte-Carlo simulation algorithm. The algorithm should be capable of generating independent realizations (each starting with a different random number) of the calling process and thereby outputting a sample of size of random variable .

Simulate the calling process = 1000 times.

Estimate from the generated sample of : (i) the mean; (ii) the first quartile, the median, the third quartile; (iii) the probabilities of events 15, 20, 30, > 40, > 5, > 6, > 7, where 5, 6, 7 are the values you choose in order to depict well the right tail of the cumulative distribution function of .