Essentials Of Stochastic Processes 2nd Edition Richard Durrett - Solutions

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(General birth and death chains). The state space is f0; 1; 2; : : :g and the transition probability has p.x; x C 1/ D px p.x; x [1] 1/ D qx forx > 0 p.x; x/ D rx for x 0 while the other p.x; y/ D 0. Let Vy D minfn 0 W Xn D yg be the time of the first visit to y and let hN .x/ D Px.VN < V0/. By
(Algorithmic efficiency). The simplex method minimizes linear functions by moving between extreme points of a polyhedral region so that each transition decreases the objective function. Suppose there are n extreme points and they are numbered in increasing order of their values. Consider the Markov
(Coupon collector’s problem). We are interested now in the time it takes to collect a set of N baseball cards. Let Tk be the number of cards we have to buy before we have k that are distinct. Clearly, T1 D 1. A little more thought reveals that if each time we get a card chosen at random from all
(Sucker bet). Consider the following gambling game. Player 1 picks a three coin pattern (for example HTH) and player 2 picks another (say THH). A coin is flipped repeatedly and outcomes are recorded until one of the two patterns appears.Somewhat surprisingly player 2 has a considerable advantage in
(Ehrenfest chain). Consider the Ehrenfest chain, Example 1.2, with transition probability p.i; i C 1/ D .N [1] i/=N, and p.i; i [1] 1/ D i=N for 0 i N. Let n D ExXn.(a) Show that nC1 D 1 C .1 [1] 2=N/n.(b) Use this and induction to conclude that n D N 2 C [1]1 [1]2 N n .x [1] N=2/From this we
(Wright–Fisher model). Consider the chain described in Example 1.7.p.x; y/ D N y !.[1]x /y.1 [1] [1]x/N[1]y where [1]x D .1 [1] u/x=N C v.N [1] x/=N.(a) Show that if u; v > 0, then limn!1 pn.x; y/ D .y/, where is the unique stationary distribution. There is no known formula for .y/, but you
(Bishop’s random walk). A bishop can move any number of squares diagonally.Let Xn be the sequence of squares that results if we pick one of bishop’s legal moves at random. Find(a) the stationary distribution and(b) the expected number of moves to return to corner (1,1) when we start there.
(King’s random walk). A king can move one squares horizontally, vertically, or diagonally. Let Xn be the sequence of squares that results if we pick one of king’s legal moves at random. Find(a) the stationary distribution and(b) the expected number of moves to return to corner (1,1) when we
(Random walk on a clock). Consider the numbers 1; 2; : : : 12 written around a ring as they usually are on a clock. Consider a Markov chain that at any point jumps with equal probability to the two adjacent numbers.(a) What is the expected number of steps that Xn will take to return to its starting
(Library chain). On each request the i th of n possible books is the one chosen with probability pi . To make it quicker to find the book the next time, the librarian moves the book to the left end of the shelf. Define the state at any time to be the sequence of books we see as we examine the shelf
(Bernoulli–Laplace model of diffusion). Consider two urns each of which contains m balls; b of these 2m balls are black, and the remaining 2m[1]b are white.We say that the system is in state i if the first urn contains i black balls and m [1] i white balls while the second contains b [1] i black
(Queue with impatient customers). Customers arrive at a single server at rate[1] and require an exponential amount of service with rate. Customers waiting in line are impatient and if they are not in service they will leave at rate ı independent of their position in the queue.(a) Show that for
(Kolmogorov cycle condition). Consider an irreducible Markov chain with state space S. We say that the cycle condition is satisfied if given a cycle of states x0; x1; : : : ; xn D x0 with q.xi[1]1; xi/ > 0 for 1 i n, we have n Y iD1 q.xi[1]1; xi / D n Y iD1 q.xi; xi[1]1/(a) Show that if q has a
(Detailed balance for three state chains). Consider a chain with state space f1; 2; 3g in which q.i; j / > 0 if i ¤ j and suppose that there is a stationary distribution that satisfies the detailed balance condition.(a) Let .1/ Dc. Use the detailed balance condition between 1 and 2 to find .2/
(Two queues in series). Consider a two station queueing network in which arrivals only occur at the first server and do so at rate 2. If a customer finds server 1 free he enters the system; otherwise he goes away. When a customer is done at the first server he moves on to the second server if it is
(When did the chicken cross the road?). Suppose that traffic on a road follows a Poisson process with rate [1] cars per minute. A chicken needs a gap of length at least c minutes in the traffic to cross the road. To compute the time the chicken will have to wait to cross the road, let t1; t2; t3; :
(Queues in series). Consider a k station queueing network in which arrivals to server i occur at rate [1]i and service at station i occurs at rate i . In this problem we examine the special case of the feed-forward system in which p.i; i C 1/ D pi for 1 i < k. In words the customer goes to the next
(Feed-forward queues). Consider a k station queueing network in which arrivals to server i occur at rate [1]i and service at station i occurs at rate i. We say that the queueing network is feed-forward if the probability of going from i to j < i has p.i; j / D 0. Consider a general three station
In the putback option at time 3 you can buy the stock for the lowest price seen in the past and the sell it at its current price for a profit of V3 D S3 min 0m3 Sm Suppose that the stock follows the binomial model with S0 D 4; u D 2; d D 1=2, and r D 1=4. (a) Compute the value function Vn.a/
Suppose Microsoft stock sells for 100 while Netscape sells for 50. Three possible outcomes of a court case will have the following impact on the two stocks.Microsoft Netscape 1 (win) 120 30 2 (draw) 110 55 3 (lose) 84 60 What should we be willing to pay for an option to buy Netscape for 50 after
The Cornell hockey team is playing a game against Harvard that it will either win, lose, or draw. A gambler offers you the following three payoffs, each for a $1 bet win lose draw Bet 1 0 1 1.5 Bet 2 2 2 0 Bet 3 0.5 1.5 0(a) Assume you are able to buy any amounts (even negative) of these bets. Is
It was crucial for our no arbitrage computations that there were only two possible values of the stock. Suppose that a stock is now at 100, but in 1 month may be at 130, 110 or 80 in outcomes that we call 1, 2 and 3. (a) Find all the (nonnegative)probabilities p1; p2 and p3 D 1 p1 p2 that make
GI=G=1 queue. Let 1; 2; : : : be independent with distribution F and Let 1; 2; : : : be independent with distribution G. Define a Markov chain by XnC1 D .Xn C n nC1/C where yC D maxfy; 0g. Here Xn is the workload in the queue at the time of arrival of the nth customer, not counting the service
Lyapunov functions. Let Xn be an irreducible Markov chain with state space f0; 1; 2; : : :g and let 0 be a function with limx!1 .x/D1, and Ex.X1/ .x/ when x K. Then Xn is recurrent. This abstract result is often useful for proving recurrence in many chains that come up in applications
Expectations of hitting times. Consider a Markov chain state space S. Let A S and suppose that C D S A is a finite set. Let VA D minfn 0 W Xn 2 Ag be the time of the first visit to A. Suppose that g.x/ D 0 for x 2 A, while for x 2 C we have g.x/ D 1 C Xy p.x; y/g.y/(a) Show that g.XVA^n/ C
Hitting probabilities. Consider a Markov chain with finite state space S. Let a and b be two points in S, let D Va ^ Vb, and let C D S fa; bg. Suppose h.a/ D 1; h.b/ D 0, and for x 2 C we have h.x/ D Xy p.x; y/h.y/(a) Show that h.Xn/ is a martingale. (b) Conclude that if Px. < 1/ > 0 for all x
Let Zn be a branching process with offspring distribution pk with p0 > 0and D Pk kpk > 1. Let ./ D P1 kD0 pkk. (a) Show that E.ZnC1 jZn/ D ./Zn .(b) Let be the solution
A branching process can be turned into a random walk if we only allow one individual to die and be replaced by its offspring on each step. If the offspring distributions is pk and the generating function is then the random walk increments have P.Xi D k1/ D pk. Let Sk D 1CX1C: : :CXn and T0 D
Consider a favorable game in which the payoffs are 1, 1, or 2 with probability 1/3 each. Use the results of Example 5.12 to compute the probability we ever go broke (i.e., our winnings Wn reach $0) when we start with $i .
Generating function of the time of gambler’s ruin. Continue with the set-up of the previous problem. (a) Use the exponential martingale and our stopping Theorem to conclude that if 0, then ex D Ex../V0 /. (b) Let 0 < s < 1.Solve the equation ./ D 1=s, then use (a) to conclude Ex.sV0 / D
Variance of the time of gambler’s ruin. Let 1; 2; : : : be independent with P. i D 1/ D p and P. i D 1/ D q D 1 p where p < 1=2. Let Sn D S0 C 1 C C n. In Example 4.3 we showed that if V0 D minfn 0 W Sn D 0g then ExV0 D x=.12p/. The aim of this problem is to compute the variance of
Mean time to gambler’s ruin. Let Sn D S0CX1C CXn where X1;X2; : : :are independent with P.Xi D 1/ D p < 1=2 and P.Xi D 1/ D 1 p. Let V0 D minfn 0 W Sn D 0g. Use Wald’s equation to conclude that if x > 0 then ExV0 D x=.1 2p/.
Let Sn D X1 C C Xn where the Xi are independent with EXi D 0 and var .Xi / D 2. (a) Show that S2 n n2 is a martingale. (b) Let D min fn W jSnj > ag. Use Theorem 5.13 to show that E a2=2. For simple random walk 2 D 1 and we have equality.
General birth and death chains. The state space is f0; 1; 2; : : :g and the transition probability has p.x; x C 1/ D px p.x; x 1/ D qx forx > 0 p.x; x/ D 1 px qx for x 0 while the other p.x; y/ D 0. Let Vy D minfn 0 W Xn D yg be the time of the first visit to y and let hN .x/ D Px.VN <
An unfair fair game. Define random variables recursively by Y0 D 1 and for n 1; Yn is chosen uniformly on .0; Yn1/. If we let U1; U2; : : : be uniform on.0; 1/, then we can write this sequence as Yn D UnUn1 U0. (a)Use Example 5.5 to conclude that Mn D 2nYn is a martingale. (b) Use the fact
Suppose that in Polya’s urn there are r red balls and g green balls at time 0.show that X D limn!1 Xn has a beta distribution.g C r 1/Š.g 1/Š.r 1/Šxg1.1 x/r1
Suppose that in Polya’s urn there is one ball of each color at time 0. Let Xn be the fraction of red balls at time n. Use Theorem 5.13 to conclude that P.Xn 0:9 for some n/ 5=9.
Lognormal stock prices. Consider the special case of Example 5.5 in which Xi D ei where i D normal.; 2/. For what values of and is Mn D M0 X1 Xn a martingale?
Let Xn be the Wright–Fisher model with no mutation defined in Example 1.9.(a) Show that Xn is a martingale and use Theorem 5.14 to conclude that Px.VN
Four children are playing two video games. The first game, which takes an average of 4min to play, is not very exciting, so when a child completes a turn on it they always stand in line to play the other one. The second one, which takes an average of 8 min, is more interesting so when they are done
At a local grocery store there are queues for service at the fish counter (1), meat counter (2), and caf´e (3). For i D 1; 2; 3 customers arrive from outside the system to station i at rate i , and receive service at rate 4 C i . A customer leaving station i goes to j with probabilities p.i; j /
At present the Economics department and the Sociology department each have one typist who can type 25 letters a day. Economics requires an average of 20 letters per day, while Sociology requires only average of 15. Assuming Poisson arrival and exponentially distributed typing times find (a) the
Customers arrive at a two-server station according to a Poisson process with rate . Upon arriving they join a single queue to wait for the next available server.Suppose that the service times of the two servers are exponential with rates a andb and that a customer who arrives to find the system
Customers arrive at a carnival ride at rate . The ride takes an exponential amount of time with rate , but when it is in use, the ride is subject to breakdowns at rate ˛. When a breakdown occurs all of the people leave since they know that the time to fix a breakdown is exponentially distributed
Customers arrive at the Shortstop convenience store at a rate of 20 per hour.When two or fewer customers are present in the checkout line, a single clerk works and the service time is 3min. However, when there are three or more customers are present, an assistant comes over to bag up the groceries
Consider a taxi station at an airport where taxis and (groups of) customers arrive at times of Poisson processes with rates 2 and 3 perminute. Suppose that a taxi will wait no matter how many other taxis are present. However, if an arriving person does not find a taxi waiting he leaves to find
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 takes an exponentially distributed amount of time with mean 40min to clean the bath room. The first one
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. Formulate a Markov chain with states 0 D all parts working, 1,2,3 D one part failed, and 4D two failures. Compute E0T4 to
A small company maintains a fleet of four cars to be driven by its workers on business trips. Requests to use cars are a Poisson process with rate 1.5 per day.A car is used for an exponentially distributed time with mean 2 days. Forgetting about weekends, we arrive at the following Markov chain for
Brad’s relationship with his girl friend Angelina changes between Amorous, Bickering, Confusion, and Depression according to the following transition rates when t is the time in months.A B C D A 4 3 1 0 B 4 6 2 0 C 2 3 6 1 D 0 0 2 2(a) Find the long run fraction of time he spends in these
We now take a different approach to analyzing the Duke Basketball chain, Example 4.11.0 1 2 3 0 3 2 1 0 1 0 5 5 0 2 1 0 2:5 1.5 3 6 0 0 6(a) Find g.i / D Ei.V1/ for i D 0; 2; 3. (b) Use the solution to (a) to show that the number of Duke scores (visits to state 1) by time t has N1.t/=t ! 0:6896
Consider the two queues in series in Problem 4.8. (a) Use the methods of Sect. 4.4 to compute the expected duration of a busy period. (b) calculate this from the stationary distribution.
A taxi company has three cabs. Calls come in to the dispatcher at times of a Poisson process with rate 2 per hour. Suppose that each requires an exponential amount of time with mean 20 min, and that callers will hang up if they hear there are no cabs available. (a)What is the probability all three
There are two tennis courts. Pairs of players arrive at rate 3 per hour and play for an exponentially distributed amount of time with mean 1 h. If there are already two pairs of players waiting new arrivals will leave. Find the stationary distribution for the number of courts occupied.
Consider a barbershop with one barber who can cut hair at rate 4 and three waiting chairs. Customers arrive at a rate of 5 per hour. (a) Argue that this new setup will result in fewer lost customers than the previous scheme. (b) Compute the increase in the number of customers served per hour.
Consider a barbershop with two barbers and two waiting chairs. Customers arrive at a rate of 5 per hour. Customers arriving to a fully occupied shop leave without being served. Find the stationary distribution for the number of customers in the shop, assuming that the service rate for each barber
Customers arrive at a full-service one-pump gas station at rate of 20 cars per hour. However, customers will go to another station if there are at least two cars in the station, i.e., one being served and one waiting. Suppose that the service time for customers is exponential with mean 6min. (a)
A computer lab has three laser printers and five toner cartridges. Each machine requires one toner cartridges which lasts for an exponentially distributed amount of time with mean 6 days. When a toner cartridge is empty it is sent to a repairman who takes an exponential amount of time with mean 1
A computer lab has three laser printers that are hooked to the network.A working printer will function for an exponential amount of time with mean 20 days. Upon failure it is immediately sent to the repair facility. There machines are worked on by two repairman who can each repair one printer in an
A computer lab has three laser printers, two that are hooked to the network and one that is used as a spare. A working printer will function for an exponential amount of time with mean 20 days. Upon failure it is immediately sent to the repair facility and replaced by another machine if there is
There are 15 lily pads and six frogs. Each frog at rate 1 gets the urge to jump and when it does, it moves to one of the nine vacant pads chosen at random. Find the stationary distribution for the set of occupied lily pads.
Three frogs are playing near a pond. When they are in the sun they get too hot and jump in the lake at rate 1. When they are in the lake they get too cold and jump onto the land at rate 2. Let Xt be the number of frogs in the sun at time t .(a) Find the stationary distribution for Xt . (b) Check
Solve the previous problem in the concrete case 1 D 1=24; 2 D 1=30; 3 D 1=84;1 D 1=3;2 D 1=5, and 3 D 1=7.
A machine is subject to failures of types i D 1; 2; 3 at rates i and a failure of type i takes an exponential amount of time with rate i to repair. Formulate a Markov chain model with state space f0; 1; 2; 3g and find its stationary distribution.
A hemoglobin molecule can carry one oxygen or one carbon monoxide molecule. Suppose that the two types of gases arrive at rates 1 and 2 and attach for an exponential amount of time with rates 3 and 4, respectively. Formulate a Markov chainmodelwith state space fC; 0; g whereCdenotes an attached
Two people who prepare tax forms are working in a store at a local mall. Each has a chair next to his desk where customers can sit and be served. In addition there is one chair where customers can sit and wait. Customers arrive at rate but will go away if there is already someone sitting in the
(a) Consider the special case of the previous problem in which 1 D 2 D 1, and 1 D 2 D 3, and find the stationary probabilities. (b) Suppose they upgrade their telephone system so that a call to one line that is busy is forwarded to the other phone and lost if that phone is busy. Find the new
Two people are working in a small office selling shares in a mutual fund.Each is either on the phone or not. Suppose that salesman i is on the phone for an exponential amount of time with rate i and then off the phone for an exponential amount of time with rate i . (a) Formulate a Markov chain
Consider the set-up of the previous problem but now suppose machine 1 is much more important than 2, so the repairman will always service 1 if it is broken.(a) Formulate aMarkov chain model for the this system with state space f0; 1; 2; 12g where the numbers indicate the machines that are broken at
Consider two machines that are maintained by a single repairman. Machine i functions for an exponentially distributed amount of time with rate i before it fails.The repair times for each unit are exponential with rate i . They are repaired in the order in which they fail. (a) Formulate a Markov
A small computer store has room to display up to three computers for sale.Customers come at times of a Poisson process with rate 2 per week to buy a computer and will buy one if at least one is available. When the store has only one computer left it places an order for two more computers. The order
A salesman flies around between Atlanta, Boston, and Chicago as follows.A B C A 4 2 2 B 3 4 1 C 5 0 5(a) Find the limiting fraction of time she spends in each city. (b) What is her average number of trips each year from Boston to Atlanta?
Suppose that the limiting age distribution in (3.9) is the same as the original distribution. Conclude that F.x/ D 1 ex for some > 0.
While visiting Haifa, Sid Resnick discovered that people who wish to travel from the port area up the mountain frequently take a shared taxi known as a sherut.The capacity of each car is five people. Potential customers arrive according to a Poisson process with rate . As soon as five people are
Each time the frozen yogurtmachine at the mall breaks down, it is replaced by a new one of the same type. (a) What is the limiting age distribution for the machine in use if the lifetime of a machine has a gamma (2,) distribution, i.e., the sum of 2 exponentials with mean 1=. (b) Find the answer
The city of Ithaca, New York, allows for 2-h parking in all downtown spaces.Methodical parking officials patrol the downtown area, passing the same point every 2 h. When an official encounters a car, he marks it with chalk. If the car is still there 2 h later, a ticket is written. Suppose that you
Show that chain in Exercise 1.38 with transition probability is 1 2 3 4 1 1=2 1=2 0 0 2 2=3 0 1=3 0 3 3=4 0 0 1=4 4 1 0 0 0 is a special case of the age chain. Use this observation and the previous exercise to compute the stationary distribution.
Consider the discrete renewal process with fj D P.t1 D j/ and Fi D P.t1 > i/. (a) Show that the age chain has transition probability q.j; j C 1/ D FjC1 Fj q.j; 0/ D 1 FjC1 Fj D fjC1 Fj for j 0(b) Show that if Et1 < 1, the stationary distribution .i/ D P.t1 > i/=Et1. (c)Let p.i; j / be the
A scientist has a machine for measuring ozone in the atmosphere that is located in the mountains just north of Los Angeles. At times of a Poisson process with rate 1, storms or animals disturb the equipment so that it can no longer collect data. The scientist comes every L units of time to check
People arrive at a college admissions office at rate 1 per minute. When k people have arrive a tour starts. Student tour guides are paid $20 for each tour they conduct. The college estimates that it loses 10 cents in good will for each minute a person waits. What is the optimal tour group size?
A machine tool wears over time and may fail. The failure time measured in months has density fT .t / D 2t=900 for 0 t 30 and 0 otherwise. If the tool fails it must be replaced immediately at a cost of $1,200. If it is replaced prior to failure the cost is only $300. Consider a replacement
Consider the set-up of Example 3.4 but now suppose that the car’s lifetime h.t/ D et . Show that for any A and B the optimal time T D 1. Can you give a simple verbal explanation?
Random Investment. An investor has $100,000. If the current interest rate is i% (compounded continuously so that the grow per year is exp.i=100/), he invests his money in a i year CD, takes the profits and then reinvests the $100,000. Suppose that the kth investment leads to an interest rateXk
In the Duke versusMiami football game, possessions alternate between Duke who has the ball for an average of 2 min and Miami who has the ball for an average of 6min. (a) In the long run what fraction of time does Duke have the ball? (b)Suppose that on each possession Duke scores a touchdown with
A worker has a number of machines to repair. Each time a repair is completed a new one is begun. Each repair independently takes an exponential amount of time with rate to complete. However, independent of this, mistakes occur according to a Poisson process with rate . Whenever a mistake occurs,
A young doctor is working at night in an emergency room. Emergencies come in at times of a Poisson process with rate 0.5 per hour. The doctor can only get to sleep when it has been 36 min (0.6 h) since the last emergency. For example, if there is an emergency at 1:00 and a second one at 1:17 then
One of the difficulties about probability is realizing when two different looking problems are the same, in this case dealing cocaine and fighting fires. In Problem 2.26, calls to a fire station arrive according to a Poisson process with rate 0.5 per hour, and the time required to respond to a
A cocaine dealer is standing on a street corner. Customers arrive at times of a Poisson process with rate . The customer and the dealer then disappear from the street for an amount of time with distribution G while the transaction is completed.Customers that arrive during this time go away never
Counter processes. As in Example 1.5, we suppose that arrivals at a counter come at times of a Poisson process with rate . An arriving particle that finds the counter free gets registered and then locks the counter for an amount of time .Particles that arrive while the counter is locked have no
A policeman cruises (on average) approximately 10 min before stopping a car for speeding. Ninety percent of the cars stopped are given speeding tickets with an$80 fine. It takes the policeman an average of 5min to write such a ticket. The other 10% of the stops are for more serious offenses,
Three children take turns shooting a ball at a basket. They each shoot until they misses and then it is next child’s turn. Suppose that child i makes a basket with probability pi and that successive trials are independent. (a) Determine the proportion of time in the long run that each child
In front of terminal C at the Chicago airport is an area where hotel shuttle vans park. Customers arrive at times of a Poisson process with rate 10 per hour looking for transportation to the Hilton hotel nearby. When seven people are in the van it leaves for the 36-min round trip to the hotel.
The times between the arrivals of customers at a taxi stand are independent and have a distribution F with mean F . Assume an unlimited supply of cabs, such as might occur at an airport. Suppose that each customer pays a random fare with distribution G and mean G. Let W.t/ be the total fares paid
Thousands of people are going to a Grateful dead concert in Pauley Pavillion at UCLA. They park their the foot cars on several of the long streets near the arena.There are no lines to tell the drivers where to park, so they park at random locations, and end up leaving spacings between the cars that
The weather in a certain locale consists of alternating wet and dry spells.Suppose that the number of days in each rainy spell is a Poisson distribution with mean 2, and that a dry spell follows a geometric distribution with mean 7. Assume that the successive durations of rainy and dry spells are
Suppose that the number of calls per hour to an answering service follows a Poisson process with rate 4. Suppose that 3/4’s of the calls are made by men, 1/4 by women, and the sex of the caller is independent of the time of the call. (a) What is the probability that in 1 h exactly two men and
Customers arrive at a bank according to a Poisson process with rate 10 per hour. Given that two customers arrived in the first 5min, what is the probability that(a) both arrived in the first 2min. (b) at least one arrived in the first 2min.
Policy holders of an insurance company have accidents at times of a Poisson processwith rate . The distribution of the time R until a claim is reported is random with P.R r/ D G.r/ and ER D . (a) Find the distribution of the number of unreported claims. (b) Suppose each claim has mean and
Ignoring the fact that the bar exam is only given twice a year, let us suppose that new lawyers arrive in Los Angeles according to a Poisson process with mean 300 per year. Suppose that each lawyer independently practices for an amount of time T with a distribution function F.t/ D P.T t/ that has
Calls originate from Dryden according to a rate 12 Poisson process.Three-fourth are local and one-fourth are long distance. Local calls last an average of 10 min, while long distance calls last an average of 5 min. Let M be the number of local calls and N the number of long distance calls in

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Essentials Of Stochastic Processes 2nd Edition Richard Durrett - Solutions

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