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probability and stochastic modeling

Questions and Answers of Probability And Stochastic Modeling

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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 +
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
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 −
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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.
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
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
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
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
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
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)
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
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
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.
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
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)
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
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
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
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
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.
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
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
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
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
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
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
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)
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
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
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.
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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