Questions and Answers of Essentials Of Stochastic Processes

Starting at some fixed time, which we will call 0 for convenience, satellites are launched at times of a Poisson process with rate . After an independent amount of time having distribution function
A light bulb has a lifetime that is exponential with a mean of 200 days.When it burns out a janitor replaces it immediately. In addition there is a handyman who comes at times of a Poisson process at
Two copy editors read a 300-page manuscript. The first found 100 typos, the second found 120, and their lists contain 80 errors in common. Suppose that the author’s typos follow a Poisson process
A copy editor reads a 200-page manuscript, finding 108 typos. Suppose that the author’s typos follow a Poisson process with some unknown rate per page, while from long experience we know that the
Wayne Gretsky scored a Poisson mean six number of points per game. sixty percent of these were goals and forty percent were assists (each is worth one point).Suppose he is paid a bonus of 3K for a
When a power surge occurs on an electrical line, it can damage a computer without a surge protector. There are three types of surges: “small” surges occur at rate 8 per day and damage a computer
Trucks and cars on highway US 421 are Poisson processes with rate 40 and 100 per hour respectively. One-eight of the trucks and one-tenth of the cars get off on exit 257 to go to the Bojangle’s in
A policewoman on the evening shift writes a Poisson mean six number of tickets per hour. Two-third’s of these are for speeding and cost $100. One-third’s of these are for DWI and cost $400. (a)
Signals are transmitted according to a Poisson process with rate . Each signal is successfully transmitted with probability p and lost with probability 1p.The fates of different signals are
Ellen catches fish at times of a Poisson process with rate 2 per hour. Forty percent of the fish are salmon, while 60%of the fish are trout.What is the probability she will catch exactly one salmon
Rock concert tickets are sold at a ticket counter. Females and males arrive at times of independent Poisson processes with rates 30 and 20 customers per hour.(a) What is the probability the first
Traffic on Snyder Hill Road in Ithaca, NY, follows a Poisson process with rate 2/3’s of a vehicle per minute. Ten percent of the vehicles are trucks, the other 90%are cars. (a) What is the
Let t1; t2; : : : be independent exponential() random variables and let N be an independent random variable with P.N D n/ D .1p/n1.What is the distribution of the random sum T D t1 C CtN
Messages arrive to be transmitted across the internet at times of a Poisson process with rate . Let Yi be the size of the i th message, measured in bytes, and let g.z/ D EzYi be the generating
Let St be the price of stock at time t and suppose that at times of a Poisson process with rate the price is multiplied by a random variable Xi > 0 with mean and variance 2. That is, St D S0 NY.t
As a community service members of the Mu Alpha Theta fraternity are going to pick up cans from along a roadway. A Poisson mean 60 members show up for work. Two-third of the workers are enthusiastic
Customers arrive at an automated teller machine at the times of a Poisson process with rate of 10 per hour. Suppose that the amount of money withdrawn on each transaction has a mean of $30 and a
An insurance company pays out claims at times of a Poisson process with rate 4 per week.Writing K as shorthand for “thousands of dollars,” suppose that the mean payment is 10K and the standard
Edwin catches trout at times of a Poisson process with rate 3 per hour.Suppose that the trout weigh an average of 4 pounds with a standard deviation of 2 pounds. Find the mean and standard deviation
Let T be exponentially distributed with rate .(a) Use the definition of conditional expectation to compute E.T jT c/
Customers arrive at a sporting goods store at rate 10 per hour. Sixty percent of the customers are men and forty percent are women. Women spend an amount of time shopping that is uniformly
Customers arrive according to a Poisson process of rate per hour. Joe does not want to stay until the store closes at T D10 PM, so he decides to close up when the first customer after time T s
The number of hours between successive trains is T which is uniformly distributed between 1 and 2. Passengers arrive at the station according to a Poisson process with rate 24 per hour. Let X denote
A math professor waits at the bus stop at the Mittag-Leffler Institute in the suburbs of Stockholm, Sweden. Since he has forgotten to find out about the bus schedule, his waiting time until the next
Calls to the Dryden fire department arrive according to a Poisson process with rate 0.5 per hour. Suppose that the time required to respond to a call, return to the station, and get ready to respond
Traffic on Rosedale Road in Princeton, NJ, follows a Poisson process with rate 6 cars per minute. A deer runs out of the woods and tries to cross the road.If there is a car passing in the next 5 s
Suppose that the number of calls per hour to an answering service follows a Poisson process with rate 4. (a) What is the probability that fewer (i.e.,
Customers arrive at a shipping office at times of a Poisson process with rate 3 per hour. (a) The office was supposed to open at 8 AM but the clerk Oscar overslept and came in at 10 AM. What is the
Suppose N.t/ is a Poisson process with rate 3. Let Tn denote the time of the nth arrival. Find (a) E.T12/, (b) E.T12jN.2/ D 5/, (c) E.N.5/jN.2/ D 5/.
Suppose 1% of a certain brand of Christmas lights is defective. Use the Poisson approximation to compute the probability that in a box of 25 there will be at most one defective bulb.Poisson
The probability of a three of a kind in poker is approximately 1/50. Use the Poisson approximation to estimate the probability you will get at least one three of a kind if you play 20 hands of poker.
Compare the Poisson approximation with the exact binomial probabilities of no success when (a) n D 10; p D 0:1, (b) n D 50; p D 0:02.
Compare the Poisson approximation with the exact binomial probabilities of 1 success when n D 20; p D 0:1.
Let Ti ; i D 1; 2; 3 be independent exponentials with rate i . (a) Show that for any numbers t1; t2; t3 maxft1; t2; t3g D t1 C t2 C t3 minft1; t2g minft1; t3g minft2; t3g C minft1; t2; t3g(b)
Ron, Sue, and Ted arrive at the beginning of a professor’s office hours. The amount of time they will stay is exponentially distributed with means of 1, 1/2, and 1/3 h. (a) What is the expected
Excited by the recent warm weather Jill and Kelly are doing spring cleaning at their apartment. Jill takes an exponentially distributed amount of time with mean 30 min to clean the kitchen. Kelly
A submarine has three navigational devices but can remain at sea if at least two are working. Suppose that the failure times are exponential with means 1,1.5, and 3 years.What is the average length
A machine has two critically important parts and is subject to three different types of shocks. Shocks of type i occur at times of a Poisson process with rate i .Shocks of types 1 break part 1,
A flashlight needs two batteries to be operational.You start with four batteries numbered 1–4. Whenever a battery fails it is replaced by the lowest-numbered working battery. Suppose that battery
Consider the set-up of the previous problem but now suppose that the two tellers have exponential service times with rates . Again, answer questions(a), (b), and (c).
Consider a bank with two tellers. Three people, Alice, Betty, and Carol enter the bank at almost the same time and in that order. Alice and Betty go directly into service while Carol waits for the
In a hardware store you must first go to server 1 to get your goods and then go to a server 2 to pay for them. Suppose that the times for the two activities are exponentially distributed with means 6
Let S and T be exponentially distributed with rates and . Let U D minfS; T g; V D maxfS; T g, and W D V U. Find the variances of U; V , and W .
Let S and T be exponentially distributed with rates and . Let U D minfS; T g and V D maxfS; T g. Find (a) EU. (b) E.V U/, (c) EV . (d) Use the identity V D S C T U to get a different looking
Alice and Betty enter a beauty parlor simultaneously, Alice to get a manicure and Betty to get a haircut. Suppose the time for a manicure (haircut) is exponentially distributed with mean 20 (30) min.
Three people are fishing and each catches fish at rate 2 per hour. How long do we have to wait until everyone has caught at least one fish?
Copy machine 1 is in use now. Machine 2 will be turned on at time t . Suppose that the machines fail at rate i . What is the probability that machine 2 is the first to fail?
A doctor has appointments at 9 and 9:30. The amount of time each appointment lasts is exponential with mean 30. What is the expected amount of time after 9:30 until the second patient has completed
The lifetime of a radio is exponentially distributed with mean 5 years. If Ted buys a 7 year-old radio, what is the probability it will be working 3 years later?
Suppose that the time to repair a machine is exponentially distributed random variable with mean 2. (a) What is the probability the repair takes more than 2 h.(b) What is the probability that the
Consider a branching process as defined in Example 1.8, in which each family has a number of children that follows a shifted geometric distribution: pk D p.1 p/k for k 0, which counts the number
Consider a branching process as defined in Example 1.8, in which each family has exactly three children, but invert Galton and Watson’s original motivation and ignore male children. In this model a
The opposite of the aging chain is the renewal chain with state space f0; 1; 2; : : :g in which p.i; i 1/ D 1 when i > 0. The only nontrivial part of the transition probability is p.0; i/ D pi .
Consider the aging chain on f0; 1; 2; : : :g in which for any n 0 the individual gets 1 day older from n to nC1 with probability pn but dies and returns to age 0 with probability 1 pn. Find
Consider the Markov chain with state space f1; 2; : : :g and transition probability p.m;m C 1/ D m=.2mC 2/ for m 1 p.m;m 1/ D 1=2 for m 2 p.m;m/ D 1=.2mC 2/ for m 2 and p.1; 1/ D 1 p.1; 2/
Consider the Markov chain with state space f0; 1; 2; : : :g and transition probability p.m;m C 1/ D 121 1 m C 2 for m 0 p.m;m 1/ D 121 C 1m C 2 for m 1 and p.0; 0/ D 1 p.0; 1/ D 3=4.
To see what the conditions in the last problem say we will now consider some concrete examples. Let px D 1=2; qx D ecx˛=2; rx D 1=2 qx for x 1 and p0 D 1. For large x; qx.1 cx˛/=2, but the
Roll a fair die repeatedly and let Y1; Y2; : : : be the resulting numbers. Let Xn D jfY1; Y2; : : : ; Yngj be the number of values we have seen in the first n rolls for n 1 and set X0 D 0. Xn is a
Customers shift between variable rate loans (V), 30 year fixed-rate loans(30), 15 year fixed-rate loans (15), or enter the states paid in full (P), or foreclosed according to the following transition
At a manufacturing plant, employees are classified as trainee (R), technician(T) or supervisor (S). Writing Q for an employee who quits we model their progress through the ranks as a Markov chain
At a nationwide travel agency, newly hired employees are classified as beginners (B). Every 6 months the performance of each agent is reviewed. Past records indicate that transitions through the
The Megasoft company gives each of its employees the title of programmer(P) or project manager (M). In any given year 70% of programmers remain in that position 20% are promoted to project manager
At the New York State Fair in Syracuse, Larry encounters a carnival game where for $1 he may buy a single coupon allowing him to play a guessing game. On each play, Larry has an even chance of
Use the second solution in Example 1.48 to compute the expected waiting times for the patterns HHH;HHT;HT T, and HTH. Which pattern has the longest waiting time? Which ones achieve the minimum value
Six children (Dick, Helen, Joni,Mark, Sam, and Tony) play catch. If Dick has the ball he is equally likely to throw it to Helen, Mark, Sam, and Tony. If Helen has the ball she is equally likely to
A warehouse has a capacity to hold four items. If the warehouse is neither full nor empty, the number of items in the warehouse changes whenever a new item is produced or an item is sold. Suppose
A bank classifies loans as paid in full (F), in good standing (G), in arrears (A), or as a bad debt (B). Loans move between the categories according to the following transition probability:F G A B F
TheMarkov chain associated with a manufacturing process may be described as follows: A part to be manufactured will begin the process by entering step 1.After step 1, 20% of the parts must be
Prove that if pij > 0 for all i and j then a necessary and sufficient condition for the existence of a reversible stationary distribution is pijpjkpki D pikpkjpj i for all i; j; k Hint: fix i and
Consider a general chain with state space S D f1; 2g and write the transition probability as 1 2 1 1 a a 2 b 1 b Use the Markov property to show that P.XnC1 D 1/ b a C b D .1 a b/P.Xn D 1/ b
At the beginning of each day, a piece of equipment is inspected to determine its working condition, which is classified as state 1 D new, 2, 3, or 4 D broken.We assume the state is a Markov chain
At the end of a month, a large retail store classifies each of its customer’s accounts according to current (0), 30–60 days overdue (1), 60–90 days overdue (2), more than 90 days (3). Their
An auto insurance company classifies its customers in three categories: poor, satisfactory and excellent. No one moves from poor to excellent or from excellent to poor in 1 year.P S E P 0:6 0:4 0 S
In a particular county voters declare themselves as members of the Republican, Democrat, or Green party. No voters change directly from the Republican to Green party or vice versa. Other transitions
Let Xn be the number of days since David last shaved, calculated at 7:30AM when he is trying to decide if he wants to shave today. Suppose that Xn is a Markov chain with transition matrix 1 2 3 4 1
An individual has three umbrellas, some at her office, and some at home. If she is leaving home in the morning (or leaving work at night) and it is raining, she will take an umbrella, if one is
A professor has two light bulbs in his garage. When both are burned out, they are replaced, and the next day starts with two working light bulbs. Suppose that when both are working, one of the two
(a) Three telephone companies A;B, and C compete for customers. Each year customers switch between companies according the following transition probability A B C A 0:75 0:05 0:20 B 0:15 0:65 0:20 C
In a large metropolitan area, commuters either drive alone (A), carpool (C), or take public transportation (T).A study showed that transportation changes according to the following matrix:A C T A 0:8
A sociologist studying living patterns in a certain region determines that the pattern of movement between urban (U), suburban (S), and rural areas (R) is given by the following transition matrix.U S
The weather in a certain town is classified as rainy, cloudy, or sunny and changes according to the following transition probability is R C S R 1=2 1=4 1=4 C 1=4 1=2 1=4 S 1=2 1=2 0 In the long run
A plant species has red, pink, or white flowers according to the genotypes RR, RW, and WW, respectively. If each of these genotypes is crossed with a pink (RW )plant then the offspring fractions are
The liberal town of Ithaca has a “free bikes for the people program.” You can pick up bikes at the library (L), the coffee shop (C) or the cooperative grocery store(G). The director of the
Bob eats lunch at the campus food court every week day. He either eats Chinese food, Quesadila, or Salad. His transition matrix is C Q S C 0:15 0:6 0:25 Q 0:4 0:1 0:5 S 0:1 0:3 0:6 He had Chinese
A midwestern university has three types of health plans: a health maintenance organization (HMO), a preferred provider organization (PPO), and a traditional fee for service plan (FFS). Experience
(a) Suppose brands A and B have consumer loyalties of 0.7 and 0.8, meaning that a customer who buys A 1 week will with probability 0.7 buy it again the next week, or try the other brand with 0.3.
Folk wisdom holds that in Ithaca in the summer it rains 1/3 of the time, but a rainy day is followed by a second one with probability 1/2. Suppose that Ithaca weather is a Markov chain. What is its
When a basketball playermakes a shot then he tries a harder shot the next time and hits (H) with probability 0.4, misses (M) with probability 0.6. When he misses he is more conservative the next time
Census results reveal that in the United States 80% of the daughters of working women work and that 30% of the daughters of nonworking women work.(a) Write the transition probability for this model.
In unprofitable times corporations sometimes suspend dividend payments.Suppose that after a dividend has been paid the next one will be paid with probability 0.9, while after a dividend is suspended
In a test paper the questions are arranged so that 3/4’s of the time a True answer is followed by a True, while 2/3’s of the time a False answer is followed by a False. You are confronted with a
Three of every four trucks on the road are followed by a car, while only one of every five cars is followed by a truck. What fraction of vehicles on the road are trucks?
A regional health study indicates that from 1 year to the next, 75% percent of smokers will continue to smoke while 25% will quit. 8% of those who stopped smoking will resume smoking while 92% will
A rapid transit system has just started operating. In the first month of operation, it was found that 25% of commuters are using the system while 75%are travelling by automobile. Suppose that each
A sociology professor postulates that in each decade 8% of women in the work force leave it and 20% of the women not in it begin to work. Compare the predictions of his model with the following data
Market research suggests that in a 5 year period 8% of people with cable television will get rid of it, and 26% of those without it will sign up for it. Compare the predictions of the Markov chain
If we rearrange the matrix for the seven state chain in Example 1.14 we get 2 3 1 5 4 6 7 2 0.2 0.3 0.1 0 0.4 0 0 3 0 0.5 0 0.2 0.3 0 0 1 0 0 0.7 0.3 0 0 0 5 0 0 0.6 0.4 0 0 0 4 0 0 0 0 0.5 0.5 0 6 0
Find limn!1 pn.i; j / for p D 1 2 3 4 5 1 1 0 0 0 0 2 0 2=3 0 1=3 0 3 1=8 1=4 5=8 0 0 4 0 1=6 0 5=6 0 5 1=3 0 1=3 0 1=3 You are supposed to do this and the next problem by solving equations. However
Do the following Markov chains converge to equilibrium?.a/ 1 2 3 4 1 0 0 1 0 2 0 0 0:5 0:5 3 0:3 0:7 0 0 4 1 0 0 0.b/ 1 2 3 4 1 0 1 0 0 2 0 0 0 1 3 1 0 0 0 4 1=3 0 2=3 0.c/ 1 2 3 4 5 6 1 0 0:5 0:5 0
Consider the Markov chain with transition matrix:1 2 3 4 1 0 0 0:1 0:9 2 0 0 0:6 0:4 3 0:8 0:2 0 0 4 0:4 0:6 0 0(a) Compute p2. (b) Find the stationary distributions of p and all of the stationary

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