Stat 330 (Spring 2015): Homework 8
Due: March 27, 2015
Show all of your work, and please staple your assignment if you use more than one sheet. Write your name,
the course number and the section on every sheet. Problems marked with * will be graded and one additional
randomly chosen problem will be graded. Show all work to be considered for partial credit.
1. An auto insurance company classifies its customers in three categories: poor, satisfactory and preferred.
No one moves from poor to preferred or from preferred to poor in one year. 40% of the customers in the
poor category become satisfactory, 30% of those in the satisfactory category moves to preferred, while
10% become poor; 20% of those in the preferred category are downgraded to satisfactory. We will model
this process with a Markov chain with three states corresponding to the three categories.
(a) Write the 1-step transition matrix for the model.
(b) What is the probability that a customer in the satisfactory category remain so next year?
(c) Customers in which category are more likely to be downgraded next year? Why?
(d) What is the probability that a customer in the preferred category still remains in the same category
after 2 years?
(e) What is the probability that a customer in the preferred category remains in the same category for
2 years in a row, i.e. he was preferred in both years? (This probability is different from the one
calculated in part (d).)
(f) What is the limiting fraction of customers in each of these categories? (Find the steady-state
distribution of this Markov chain.)
2. * Jobs are sent to a server at a rate of 2 jobs per minute. We will model job arrivals using a homogenous
Poisson process.
(a) What is the probability of receiving more than 3 jobs in a period of one minute?
(b) What is the probability of receiving more than 30 jobs in a period of 10 minutes?
(c) What is the expected value and the variance of interarrival times?
(d) Compute the probability that the next job does not arrive during the next 30 seconds?
(e) Compute the probability that the time till the fourth job arrives exceeds 40 seconds?
3. Suppose that the telephone calls arriving at the switchboard of a small corporation follows a Poisson
process. Let X(t) be the number of calls received by time t that day, with X(0) = 0. Suppose that the
intensity parameter is λ = 0.5 calls per minute. Assume that the number of calls arriving during any
period has a Poisson distribution.
(a) What is the probability that no calls will arrive in a 5-minute period?
(b) What is the expected amount of time between the 10th and 11th callers?
(c) On average, how much time is needed to get their 100th caller of the day?
(d) What is the probability that the 100th caller arrives by their lunch break at t = 240?
4. (Baron’s book): 6.3
5. * (Baron’s book): 6.4
6. (Baron’s book): 6.7
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