Reference no: EM131011365

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Question 1. Let {X_n} (n ≥ 0) represent a branching process, where X_n denotes the population size in the nth generation. The initial population size is 1, i.e. X0 = 1, and in each generation the number of offspring produced by each individual that survive to the next generation has the binomial distribution with parameters 2 and 1 , so that the offspring distribution {p_n} is given by p₀ = 1/4, p₁ = 1/2, p₂ =1/4

The numbers of surviving offspring produced by different individuals are statistically independent of each other.

(i) Write down an expression for the probability generating function G(z) of the offspring distribution.

Let Gn(z) denote the probability generating function of the number of individuals in the population in the nth generation (n ≥ 1).

(ii) By conditioning on X1, prove that

Gn (z) = 1/4[1 + G_n-1(z)]²     ( n ≥ 2).

(iii) Let θ_n = P(X_n = 0) (n ≥ 1), the probability that the population has become extinct by the nth generation. Using the relationship of part (ii), find a recurrence relation for the θ_n and deduce the value of θ = lim_n→∞θ_n, the probability of ultimate extinction of the population.

(iv) Let μ_n denote the mean population size in the nth generation (n ≥ 1). By differentiating the relationship of part (ii), find a recurrence relation for the μ_n and deduce that μ_n = 1 (n ≥ 1).

(v) Let σ_n² denote the variance of the population size in the nth generation (n ≥ 1). By differentiating the relationship of part (ii) twice, find a recurrence relation for the σ_n² and deduce that

σ_n² = n/2 (n≥1)

Question 2. A Markov chain model for death from a particular infection of an animal from some population has four states as follows.

state 1: animal alive and free from infection

state 2: animal alive and infected

state 3: animal dead from the infection

state 4: animal dead from another cause

The unit of time is taken to be one year, and the transition matrix P is given by

	5/8	1/4	0	1/8
P =	1/4	1/2	1/8	1/8
	0	0	1	0
	0	0	0	1

(i) Identify, with reasons, the states that are transient and those that are recurrent.

(ii) If in a given year the animal is in state 1, calculate the probability that two years later it is in state i, for each of i = 1, 2, 3, 4.

(iii) Let xi (i = 1, 2) denote the probability that the animal eventually dies from the infection, given that it starts in state i. Write down and solve a pair of backward equations for the xi.

(iv) Let yi (i = 1, 2) denote the expected number of years until the animal dies (either from the infection or from another cause), given that it starts in state i. Write down and solve a pair of backward equations for the yi.

Question 3. In a Markov chain model for the state of a machine in a factory, X_n denotes the state in week n, where the state space is {0, 1, 2, ..., M} for some positive integer M. The state 0 represents a machine that is in perfect working order and the states 1, 2, ..., M represent successively worse states of operation. When the machine reaches state M then it is serviced so that it returns to state 0 the following week. The transition rates P_i,j of the model are specified as follows:

p_i,j+1 = θ (0 ≤ i ≤ M-1)

p_{i,i = 1}- θ (0 ≤ i ≤ M-1)

p_M,0 = 1

where is a parameter with 0 < θ < 1.

(i) Write down a set of equations satisfied by the stationary distribution {Π_i} (0 ≤ i ≤ M) and deduce expressions for the probabilities Π_i (0 ≤ i ≤ M) in terms of θ and M.

(ii) If c, K are constants such that the cost per week of running the machine is given by ci whenever the machine is in state i (0 ≤ i ≤ M) and K is the cost of servicing the machine, show that the long-term average cost per week of running and servicing the machine is given by

(1/2cM(M-1) + θ(cM+k))/(M + θ)

Question 4. Consider a continuous time Markov chain model {N(t): t ≥ 0} for a car park with M spaces, where N(t) denotes the number of spaces occupied at time t. It is assumed that cars arrive at the car park according to a Poisson process with rate and that if there are any free spaces then an arriving car is parked, but if all spaces are occupied then the arriving car leaves. It is also assumed that if at any time there are i spaces occupied then the probability that one of the parked cars leaves in the next small interval of time of length h is μih + o(h).

(i) Write down the state space for the model and specify the transition rates.

(ii) Define

p_n(t) = P(N(t) = n|N(0) = v) (0 ≤ n ≤ M)

where v represents the number of cars in the car park at time 0. Derive the forward equations for the p_n(t), distinguishing between the cases n = 0, 1 ≤ n ≤ M - 1 and n = M.

(iii) By first writing down the detailed balance equations (or otherwise), obtain the equilibrium distribution {Π_n} (0 ≤ n ≤ M) for the chain. Show in particular that the long-term proportion of time that the car park is full is given by

(ρ^M/M!)/Σ_n=0^M ρⁿ/n!, where ρ = λ/μ

Question 5. In a single-server queue, customers arrive according to a Poisson process with rate λ and service times have a gamma distribution with parameters μ and 2, so that the probability density function of the service times is

μ²te^-μt (t≥0)

(i) Find the mean of the service time distribution. Define the traffic intensity ρ of a queue and, for the present case, write down an expression for ρ in terms of λ and μ. State the condition for an equilibrium distribution to exist.

(ii) Conditional upon the length t of a service time being given, write down the distribution of the number of customers who arrive during the service time and show that the probability generating function (pgf) G(z) of this distribution is given by

G(z) = e^-λt(1-z).

(iii) Deduce that, unconditionally, the pgf K(z) of the number of customers who arrive during a service time is given by

K(z) = μ/(μ + λ- λz)

(iv) Deduce an expression for the expected number of customers who arrive during a service time and discuss how your expression relates to the results of part (i).

Question 6. Consider the AR(2) model

Y_t = 3/10 Y_t-1 + 1/10 Y_t-2 + ε_t(-∞<t<∞)

for a process {Y_t}, where {ε_t} is a white noise process.

(i) Find the roots of the autoregressive characteristic equation and verify that the stationarity condition is satisfied.

(ii) The infinite moving average representation of Y_t can be written

Y_t = Σ_t=0^∞ ψ_iε_t-i

By substituting this expression into the model equation, find a set of recurrence relations satisfied by the i together with appropriate initial conditions.

(iii) Solve these recurrence equations to show that

ψ_i = 5/7.1/2 i + 2/7 -1/5 ⁱ      (i ≥ 0)

(iv) Deduce that

Var(Y_t) = 25/22 σ²

where σ² is the variance of the white noise process {ε_t}.

Question 7. A time series consists of the mean lake surface elevation in feet at a particular location on Lake Huron in July for the 127 years from 1860 to 1986.

(a) The sample autocorrelation function (acf) and sample partial autocorrelation function (pacf) are tabulated below for the first 20 lags.

lag	acf	pacf	lag	acf	pacf
1	0.831	0.831	11	0.216	-0.012
2	0.643	-0.157	12	0.194	0.071
3	0.513	0.083	13	0.212	0.111
4	0.436	0.062	14	0.207	-0.115
5	0.403	0.097	15	0.187	0.038
6	0.364	-0.033	16	0.170	0.002
7	0.340	0.082	17	0.118	-0.107
8	0.309	-0.027	18	0.085	0.007
9	0.310	0.133	19	0.077	0.051
10	0.273	-0.141	20	0.077	0.006

(i) Explain in general how the acf and pacf may be used to determine whether an observed time series appears to be stationary and, if so, which of the family of ARMA models it might be appropriate to attempt to fit to the series.

(ii) Draw conclusions in the present case.

(b) An edited version of the output from a statistical package for fitting one of the ARMA models to the Lake Huron data is given below.

ARMA 'level'

Final Estimates of Parameters

Type	Coef	SE Coef	T	P
AR   1	0.8660	0.0470	18.43	0.000
Constant	77.6617	0.0632	1228.99	0.000
Mean	579.503	0.472

Number of observations: 127

Residuals:   SS = 62.0777 (backforecasts excluded) MS = 0.4966             DF = 125

Modified Box-Pierce (Ljung-Box) Chi-Square statistic

Lag	12	24	36	48
Chi-Square	12.9	26.4	36.3	46.9
DF	10	22	34	46
P-Value	0.229	0.236	0.364	0.434

(i) State which of the family of ARMA models has been fitted to the data and write down explicitly the equation of the fitted model, including the white noise term.

(ii) Discuss how the above output may be used to determine whether the fitted model provides an adequate fit to the data, and draw conclusions.

(iii) Given that the mean level in 1986 was 581.27, calculate a 95% prediction interval for the mean level in 1987.

Question 8. Let y₁, y₂, ..., y_t, ... denote the observed values of a time series starting from time 1.

(i) Let L_t denote the smoothed value (the level) of the series at time t obtained using simple exponential smoothing. If denotes the smoothing constant, write down the updating equation for L_t in terms of L_t-1 and y_t. State what is the corresponding forecast yˆ_t (h) at time t for lead time h (h ≥ 1).

(ii) Assume that the value L₁ is taken to be equal to y₁. By applying the updating equation iteratively, find an explicit expression, in as simple terms as you can, for L_t in terms of y₁, y₂, ..., y_t.

(iii) Denote by e_t the one-step-ahead forecast error, e_t = y_t - yˆ_t-1(1) (t ≥ 2).

Rewrite the updating equation for L_t in terms of L_t-1 and e_t.

The monthly sales of TV sets in a retail store have been recorded over 24 successive months and the method of simple exponential smoothing is applied. The sales for the first four months, t = 1, 2, 3, 4, are 30, 32, 30, 39, respectively.

(iv) Using the initial value y₁ of the series as the initial smoothed value L₁, and taking the smoothing constant to be 0.2, calculate (to 2 decimal places) the smoothed value and forecast error for t = 2, 3, 4, respectively.

In order to explore what might be the best value of the smoothing constant to use, the method of simple exponential smoothing has been applied to the whole run of the series using a variety of values of the smoothing constant. In each case the values of the mean absolute deviation (MAD) and the error sum of squares (SSE) have been calculated. The results are tabulated below.

α	MAD	SSE
0.10	3.47	383.98
0.20	3.13	335.95
0.25	3.05	333.47
0.30	3.00	336.22
0.35	3.03	342.38
0.40	3.07	351.11
0.50	3.22	374.78

(v) Define the MAD and the SSE.

(vi) Given the tabulated results, discuss what value of you would recommend.