An Introduction To Stochastic Modeling 4th Edition Mark A. Pinsky, Samuel Karlin - Solutions

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3.13 Use the grad function of Figure 3.3 to check the gradient at the maximumlikelihood estimate in Exercises 3.10 and 3.12.
3.12 Use the Matlab fminsearch routine to obtain the maximum-likelihood estimates for the Cauchy likelihood, using the function in Figure 3.8(b).
3.11 Construct a Matlab m-file to evaluate the beta-geometric negative loglikelihood. You can base this on the code in Figure 3.2(b), but note that as the function is to provide a single value of the negative log-likelihood for a given pair of scalar quantities, (m, t), you do not need the
3.10 In this question we look at the logit model for quantal assay data. At each of a number of doses, xi, groups of ni individuals are given the dose, and ri respond, 1 ≤ i ≤ k. In the logit model, the probability of response at dose xi is written as P(xi)=1/(1 + e−(α+βxi))(i) Study the
3.9 Check the Matlab function below for the simulated annealing algorithm.Modify it for use with a two-parameter Cauchy log-likelihood. Consider how you might adapt it to carry out the hybrid optimisation procedure.
3.8∗ In Section 3.3.4 we see how reflection operates for the simplex method.Write down the corresponding equations for expansion and contraction.Program the simplex method in Matlab, using the flow diagram in Bunday(1984).
3.7 Use the Matlab tic and toc commands to compare the time taken by the program of Figure 3.2(a) with that of a more standard version in which the sums are formed via a loop.
3.6 Use maximum-likelihood to fit the following model:pr(x) = βΓ(x + α)γx−1 x!Γ(1 + α) , for x ≥ 1,in which α, β and γ are the parameters to be estimated, to the word type frequencies of Table 1.8. Obtain the expected values, and comment on the quality of the fit.
3.5 Stimulus-response experiments in psychology result in data such as that illustrated below:
3.4 Consider the microbial infection data of Example 1.2. A possible model for the progress of the infection is that a single infecting bacterium living according to a stochastic linear birth-and-death process (see Cox and Miller, 1965, p.156) with respective birth and death rates λ and μ, with
3.3∗ (Brooks et al., 1997.) The data below describe the number of dead fetuses in litters of mice. Use the Matlab routine fminsearch to fit the beta binomial model, which has the probability function:pr(X = x) =*n x+ x−1 r=0 (μ + rθ)n−x−1 r=0 (1 − μ + rθ)n−1 r=0 (1 + rθ) ,= Q(x, n,
3.2∗ Use a computer algebra package such as Maple to verify the forms for A and E[A] given in Section 3.3.2 for the Cauchy likelihood of Example 3.1.
3.1 The numerical approximation to the gradient of a function f(x) for a scalar quantity x, given in Figure 3.3, is f(x), given by:2δf(x) = f(x + δ) − f(x − δ).Use Matlab to compare the performance of this method with the alternative which results in:12δf(x) = f(x − 2δ) − 8f(x −
2.27 Suggest a suitable model for the cell division time data of Table 1.15.
2.26 On each of k occasions, a closed population of wild animals, of fixed but unknown size N, is sampled at random and any animals in the sample that have not been marked previously are marked and returned to the population. The objective is to estimate N.The general form for the likelihood is
2.25 The data below describe the onset of menopause for women classified according to whether or not they smoke (Healy, 1988, p.85.)
2.24 The data below describe the mortality of adult flour beetles (Tribolium confusum) after 5 hours’ exposure to gaseous carbon disulphide (CS2);from Bliss (1935).Dose(mg/l) 49.06 52.99 56.91 60.84 64.76 68.69 72.61 76.54 No. of beetles 59 60 62 56 63 59 62 60 No. killed 6 13 18 28 52 53 61 59
2.23 For the period 1920–1979, lengths of ‘very warm spells’ (periods of three or more consecutive days with maximum temperature more than 4◦C above the long-term mean) have been recorded at Edgbaston, Birmingham. The results are as follows:
2.22 (Continuation of Exercise 2.4.) A species of wasp lays its eggs on larvae.Suppose that any larva has X encounters with wasps where X has the Poisson distribution, pr(X = k) = e−λλk k! , for k = 0, 1, 2,....Suppose that an egg is always laid at the first encounter, but that at all
2.21 When introduced into an area containing both fine and coarse sand, antlions are thought to prefer to dig burrows in fine sand, but also to avoid other ant-lions. The data below describe the results of 62 experiments in which either 3 or 4 ant-lions are introduced into an area with fine and
2.20 A study of the mortality of grey herons involves the release of a cohort of birds marked soon after hatching in a single year. The probabilities of birds being reported dead in each of the following four years are partially shown below in temporal order, for a model with a constant reporting
2.19 The number of male rabbits in 100 litters, each containing 3 rabbits, are described in the table below:Number of males in litter 0 1 2 3 Number of litters 25 39 27 9 Consider how you would model these data, and construct a likelihood.
2.18 In Section 2.3, two alternative expressions are given for the probability function of the beta-geometric distribution. Verify that they are equivalent.
2.17 For a random sample, (u1,...,un) from a U(0, θ) distribution, find the maximum-likelihood estimate of θ.
2.16 Explain why the multinomial distribution is appropriate in Section 2.2.
2.15 (Morgan, 1982.) Two further examples of data resulting from studies of polyspermy are shown below. In (a) the eggs were sea urchins Echinus esculentus, with a sperm density of 6 × 107 sperm/ml at 15◦C. In (b) the
2.14 (Ridout et al., 1999.) Figure 2.3 shows the topological branching structure of a typical strawberry inflorescence. Consider how you would form a probability model for the data below. The data describe the numbers of inflorescences having a given number of flowers of rank r classified by the
2.13 Suppose that the random variable X ∼ Po(λ) and that λ ∼ Γ(α, β), where α is a positive integer. Show that the marginal distribution of X has the negative-binomial form:pr(X = k) = k + α − 1α − 1 1(1 + β)αβk(1 + β)k , k ≥ 0 .
2.12 Suppose that the random variable X ∼ N(0, θ−1), and that θ ∼ Γ( 1 2 , 2γ ), where γ is a positive integer. Show that the marginal distribution of X is the t-distribution, tγ.
2.11 Verify the expressions for the mean and variance of the beta-geometric distribution, given in Section 2.3, using the following expressions for the conditional mean and variance of random variables:E[X] = Ep [E[X|p]] , and Var(X) = Ep[Var (X|p)] + Varp [E[X|p]] , when the subscript p denotes
2.10∗ Verify the results of Section 2.3, that under the beta-geometric model, the expected remaining waiting time for couples who have had j failed cycles is given by:E[(X − j)|X>j] = 1 − θμ − θ+ jθ(μ − θ) .Hint: First of all show that under the beta-geometric model, conditional
2.9 In this question we consider maximum-likelihood estimation for the polyspermy example under the single Poisson process model. Suppose that we write the log-likelihood as:l(λ) = C − λA + B(log λ)for positive constants A, B and C. Obtain the value of λ that results in dl dλ = 0, and show
2.8 Verify that the value of ˆp in Equation (2.1) maximises the relevant likelihood. Note that the problem here is equivalent to finding the maximumlikelihood estimate of the binomial probability p when r successes occur out of n independent Bernouilli trials. Why is this?
2.7 Derive the probability function of the beta-binomial distribution. An urn contains r red and b black balls. Balls are removed at random, one at a time, and each is replaced, together with an extra ball of the same colour as the ball drawn. Show that the number of red balls drawn after n
2.6 The data below provide the numbers of responses a set of human subjects made when hearing the consonant phonemes, p, t and k (Clarke, 1957).Write down a probability model for the data, and try to fit the model to the data using maximum likelihood.
2.5 For any pair of plants at any census point, Catchpole et al. (1987) model the ant-guard data of Table 1.10 by means of the trinomial distribution pr(n1, n2, n3) = n1 + n2 + n3 n1, n2, n3 (m1α)n1 (m2β)n2 (m3γ)n3(m1α + m2β + m3γ)n1+n2+n3 , for parameters α, β, γ, subject to the
2.4 Many species of insect parasite lay their eggs in live hosts, and try to do this to avoid superparasitism, that is, to try to avoid parasitising an already parasitised host. Consider how you would model the following data, from Simmonds (1956), which relate to the parasitic behaviour of
2.3 For the flour beetle data of Table 1.4, we may focus on the endpoint mortality data, corresponding to the observations on day 13. A logistic or logit analysis of these data is based on the model, Pr(death by day 13 on dose di) = [1 + exp{−(α1s + α2s log di)}]−1, where s = 1, 2 is the
2.2 The data below are taken from Harlap and Baras (1984). Here we have cycles to conception, following the cessation of contraception use, which was either use of the contraceptive pill or some other method. Discuss the data, and consider suitable stochastic models. Write down a likelihood.
2.1 Obtain an estimate of p for each group based on N1, the number of observations for which X = 1. What is the distribution of N1? Use this information to approximate standard errors for your estimates. What can you say about the success rates for the two groups of drivers?
1.12 The data in Table 1.15 describe cell division times for mother and daughter cells of the budding yeast, Saccharomyces cerevisiae, taken from Ridout et al. (2006). Consider whether there is a difference between mothers and daughters.
1.11 The data of Table 1.14 continue the data presented in Table 1.10. Compare the two sets of data, and consider whether there are any striking temporal effects.
1.10 In a study of the behaviour of bees, plants previously sprayed with pesticides (which are damaging to bees) were also sprayed with chemical repellents, denoted by A, B and C. These were arranged in eight dishes and counts taken of the numbers of bees arriving at the dishes at intervals of 1
1.9 Discuss possible biological reasons for the synchronisation of hatching observed in the data of Table 1.6.
1.8 The Independent newspaper published in Britain on 23 March 1999 discussed the attempts of couples trying to give birth to ‘millenium babies’ as follows: ‘The race to conceive a millenium baby is certain to end in tears,... only one in three will conceive in the next month.’ Discuss in
1.7 The scientific Journal of Agricultural, Biological and Environmental Statistics (JABES) received about 100 submitted papers a year for the period, 2003–2006. These submissions are distributed over months as shown in Table 1.12.By means of simple tabulations and plots, consider whether there
1.6 Each April, little blue penguins (Eudyptula minor ) return to Somes Island in Wellington Harbour, New Zealand, to choose a mate, build a nest and rear chicks. Pledger and Bullen (1998) proposed a test for fidelity in the choice of mates. Read the paper and critically assess the suggested test.
1.5 Locate and read at least one of the papers by Brooks et al. (1991), Goldman (1993), Jφrgensen et al. (1991), Royston (1982), and Ridout (1999), all of which provide examples of complex stochastic models and how they are fitted to data.
1.4 A simple model for the data of Table 1.5 would be one in which each year a heron has a probability, φ say, of surviving that year. Consider whether you think this would be an adequate description of the data. What else is needed in order to be able to write down probabilities of birds being
1.3 Data which might otherwise be expected to result from a Poisson distribution are sometimes found to contain too many zeroes. Suggest one possible mechanism for this ‘zero inflation,’ and consider how you might describe the data by means of a probability model. For discussion, see the paper
1.2 Try to construct a probability model for the ant-lion distribution data of Table 1.10.
1.1 Locate and read the paper by Pielou (1963b), which uses the geometric distribution to model the runs of diseased trees in forest transects.
Exercise 11.3 Let X be a real-valued random variable. Show that jX.t/j 1 and X.0/ D 1.
Exercise 11.2 Let X be a real-valued random variable. Show that the characteristic function is continuous in t.
Exercise 11.1 If X is normally distributed with mean zero and variance 1, then E.X2n/ D .2n/!=2nn!for even moments and zero for odd moments.
Exercise 10.7 Prove the necessary conditions (i’), (ii’), and (iii’).
Exercise 10.6 Suppose that we are given constants v1 < v2;q1 > 0;q2 > 0 satisfying the conditions(i)-(ii)-(iii). Then, we can define a10;a01 so that (10.33) holds.
Exercise 10.5 Prove that any PDE of the form (10.33) satisfies conditions (i)-(ii)-(iii), when one writes it in the form (10.33).
Exercise 10.4 Show that (10.32) follows from (10.31), when we take N D 2.If we write the PDE (10.31) in the formthen a20 D 1; a11 D .v1 Cv2/; a02 D v1v2; a10 D q1 Cq2; a01 D v1q2 Cv2q1; a0 D 0.Then, a short calculation shows that (i) The polynomial 2 Ca11Ca02 D 0 has two distinct real roots v1 (ii)
Exercise 10.3 Let g.x/ be a solution of the differential equation g00 C2g0 C2g D 0, where is a real parameter. Find the general solution in case .1/jj j D 1; and .3/jj > 1.In detail, we have the following results: i P11(t) e cosht 1-2+ sinht/1 H2 1- P-1(t,)=e sinht1-2 1- P-1,1(t,p)=e sinht1-2
Exercise 10.2 Define the Fourier transform of an integrable function f by Of .y/ D RR f .x/e????ix dx.Show that if f .x; t/ is a soluton of the telegraph equation, then Of D g is a solution of the ordinary differential equation gtt C2gt C2g D 0. Assume as much smoothness and decay as you need.The
Exercise 10.1 Suppose that P.t/ is the family of transition matrices that correspond to a two-state Markov chain with the values v1; v2 and P[V.t/ D v1jV.0/ D v1] D 1????tbCo.t/;P[V.t/ D v2jV.0/ D v2] D 1????taCo.t/; t # 0;where a > 0;b > 0.(a) Show that the transition matrix is given by the
9.5.1 Suppose three service stations are arranged in tandem so that the departures from one form the arrivals for the next. The arrivals to the first station are a Poisson process of rate D 10 per hour. Each station has a single server, and the three service rates are 1 D 12 per hour, 2 D 20
9.4.2 Consider the preemptive priority queue of Section 9.4.5 and suppose that the arrival rate is D 4 per hour. Two classes of customers can be identified, having mean service times of 12 min and 8 min, and it is proposed to give one of these classes priority over the other. Recommend which
9.4.1 Consider the two-server overflow queue of Section 9.4.4 and suppose the arrival rate is D 10 per hour. The two servers have rates 6 and 4 per hour. Recommend which server should be placed first. That is, choose between1 D 6;2 D 4 and1 D 4;2 D 6;and justify your answer. Be explicit about
9.4.3 Consider a two-server system in which an arriving customer enters the system if and only if a server is free. Suppose that customers arrive according to a Poisson process of rate D 10 customers per hour, and that service times are exponentially distributed. The servers have different
9.4.2 Customers arrive at a checkout station in a small grocery store according to a Poisson process of rate D 1 customer per minute. The checkout station can be operated with or without a bagger. The checkout times for customers are exponentially distributed, and with a bagger the mean checkout
9.4.1 Consider a two-server system in which an arriving customer enters the system if and only if a server is free. Suppose that customers arrive according to a Poisson process of rate D 10 customers per hour, and that service times are exponentially distributed with a mean service time of 6 min.
9.3.2 In operating a queueing system with Poisson arrivals at a rate of D 1 per unit time and a single server, you have a choice of server mechanisms. Method A has a mean service time of D 0:5 and a variance in service time of 2 D 0:2, while Method B has a mean service time of D 0:4 and a
9.3.1 Let X.t/ be the number of customers in an M=G=1 queueing system at time t, and let Y.t/ be the number of customers who have entered the system and completed service by time t. Determine the joint distribution of X.t/ and Y.t/.
9.3.5 Let X.t/ be the number of customers in an M=G=1 queueing system at time t.Suppose that X.0/ D 0. Evaluate M.t/ D E[X.t/], and show that it increases monotonically to its limiting value as t!1.
9.3.4 Customers arrive at a checkout station in a market according to a Poisson process of rate D 1 customer per minute. The checkout station can be operated with or without a bagger. The checkout times for customers are random. With a bagger the mean checkout time is 30 s, while without a bagger
9.3.3 Customers arrive at a tool crib according to a Poisson process of rate D 5 per hour. There is a single tool crib employee, and the individual service times are random with a mean service time of 10 min and a standard deviation of 4 min.In the long run, what is the mean number of workers at
9.3.1 Suppose that the service distribution in a single server queue is exponential with rate ; i.e., G./ D 1????e???? for 0. Substitute the mean and variance of this distribution into (9.35) and verify that the result agrees with that derived for the M=M=1 system in (9.12).
9.2.5 Customers arrive at a service facility according to a Poisson process having rate . There is a single server, whose service times are exponentially distributed with parameter . Let N.t/ be the number of people in the system at time t. Then, N.t/ is a birth and death process with parameters
9.2.4 The problem is to model a queueing system having finite capacity. We assume arrivals according to a Poisson process of rate , with independent exponentially distributed service times having mean 1=, a single server, and a finite system capacity N. By this we mean that if an arriving
9.2.3 Determine the stationary distribution for an M=M=2 system as a function of the traffic intensity D =2, and verify that L D W.
9.2.2 Determine the mean waiting time W for an M=M=2 system when D 2 and D 1:2. Compare this with the mean waiting time in an M=M=1 system whose arrival rate is D 1 and service rate is D 1:2. Why is there a difference when the arrival rate per server is the same in both cases?
9.2.1 Determine explicit expressions for 0 and L for the M=M=s queue when s D 2.Plot 1???? 0 and L as a function of the traffic intensity D =2.
9.1.1 Two dump trucks cycle between a gravel loader and a gravel unloader. Suppose that the travel times are insignificant relative to the load and unload times, which are exponentially distributed with parameters and , respectively. Model the system as a closed queueing network. Determine the
9.1.1 What design questions might be answered by modeling the following queueing systems?The Customer The Server(a) Arriving airplanes The runway(b) Cars A parking lot(c) Broken TVs Repairman(d) Patients Doctor(e) Fires Fire engine company What might be reasonable assumptions concerning the arrival
9.6.1 In the case m 1;n 1, verify that m;n as given following (9.84) satisfies the equation for the stationary distribution (9.81).
9.5.1 Suppose three service stations are arranged in tandem so that the departures from one form the arrivals for the next. The arrivals to the first station are a Poisson process of rate D 10 per hour. Each station has a single server, and the three service rates are 1 D 12 per hour, 2 D 20
9.4.3 Balking refers to the refusal of an arriving customer to enter the queue. Reneging refers to the departure of a customer in the queue before obtaining service.Consider an M=M=1 system with reneging such that the probability that a specified single customer in line will depart prior to service
9.4.2 Consider the preemptive priority queue of Section 9.4.5 and suppose that the arrival rate is D 4 per hour. Two classes of customers can be identified, having mean service times of 12 min and 8 min, and it is proposed to give one of these classes priority over the other. Recommend which
9.4.3 Consider a two-server system in which an arriving customer enters the system if and only if a server is free. Suppose that customers arrive according to a Poisson process of rate D 10 customers per hour, and that service times are exponentially distributed. The servers have different
9.4.2 Customers arrive at a checkout station in a small grocery store according to a Poisson process of rate D 1 customer per minute. The checkout station can be operated with or without a bagger. The checkout times for customers are exponentially distributed, and with a bagger the mean checkout
9.4.1 Consider a two-server system in which an arriving customer enters the system if and only if a server is free. Suppose that customers arrive according to a Poisson process of rate D 10 customers per hour, and that service times are exponentially distributed with a mean service time of 6 min.
9.3.2 In operating a queueing system with Poisson arrivals at a rate of D 1 per unit time and a single server, you have a choice of server mechanisms. Method A has a mean service time of D 0:5 and a variance in service time of 2 D 0:2, while Method B has a mean service time of D 0:4 and a
9.3.1 Let X.t/ be the number of customers in an M=G=1 queueing system at time t, and let Y.t/ be the number of customers who have entered the system and completed service by time t. Determine the joint distribution of X.t/ and Y.t/.
9.3.5 Let X.t/ be the number of customers in an M=G=1 queueing system at time t.Suppose that X.0/ D 0. Evaluate M.t/ D E[X.t/], and show that it increases monotonically to its limiting value as t!1.
9.3.4 Customers arrive at a checkout station in a market according to a Poisson process of rate D 1 customer per minute. The checkout station can be operated with or without a bagger. The checkout times for customers are random. With a bagger the mean checkout time is 30 s, while without a bagger
9.3.3 Customers arrive at a tool crib according to a Poisson process of rate D 5 per hour. There is a single tool crib employee, and the individual service times are random with a mean service time of 10 min and a standard deviation of 4 min.In the long run, what is the mean number of workers at
9.3.2 Consider a single-server queueing system having Poisson arrivals at rate . Suppose that the service times have the gamma densitywhere > 0 and > 0 are fixed parameters. The mean service time is = and the variance is =2. Determine the equilibrium mean queue length L. g(y)= -- T (a) for y
9.3.1 Suppose that the service distribution in a single server queue is exponential with rate ; i.e., G./ D 1????e???? for 0. Substitute the mean and variance of this distribution into (9.35) and verify that the result agrees with that derived for the M=M=1 system in (9.12).
9.2.7 Let X.t/ be the number of customers in an M=M=1 queueing system at time t.Suppose that X.0/ D 0.(a) Derive the forward equations that are appropriate for this process by substituting the birth and death parameters into Chapter 6, equation (6.24).(b) Show that M.t/ D E[X.t/] satisfies the
9.2.5 Customers arrive at a service facility according to a Poisson process having rate . There is a single server, whose service times are exponentially distributed with parameter . Let N.t/ be the number of people in the system at time t. Then, N.t/ is a birth and death process with parameters n
9.2.3 Determine the stationary distribution for an M=M=2 system as a function of the traffic intensity D =2, and verify that L D W.
9.2.2 Determine the mean waiting time W for an M=M=2 system when D 2 and D 1:2. Compare this with the mean waiting time in an M=M=1 system whose arrival rate is D 1 and service rate is D 1:2. Why is there a difference when the arrival rate per server is the same in both cases?
9.2.1 Determine explicit expressions for 0 and L for the M=M=s queue when s D 2.Plot 1???? 0 and L as a function of the traffic intensity D =2.

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An Introduction To Stochastic Modeling 4th Edition Mark A. Pinsky, Samuel Karlin - Solutions

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