LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 43

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2008

    ST 3809 / 3806 / 3800 – STOCHASTIC PROCESSES

 

 

 

Date : 29/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Section-A (10×2=20 marks)

Answer ALL the questions. Each question carries TWO marks.

 

1. Define a) Independent increments and b) Stationary increments of a stochastic process.
2. Let {X_n, n=0,1,2,…} be a Markov chain with state space S={1,2,3} and transition probability matrix

 

 

P =       ½     ¼      ¼

2/3     0         1/3

3/5     2/5      0

Compute P[X₁=2, X₂=3, X₃=1, X₄=3│X₀=3]

1. Define: a) Recurrent state b) Ergodic Markov chain.
2. If   lim  p_jj ⁽ⁿ⁾ >0, show that state j is positive recurrent.

n→∞

1. Derive the probability generating function of Yule process corresponding to the homogeneous case, when X(0)=1.
2. Write down the postulates for a birth and death process.
3. Give an example of naturally occurring process that can be modelled as a renewal process.
4. Define a semi Markov process.
5. Given the probability generating function to be f(s)= (as+b), a+b=1, a,b>0, determine the extinction probability if X(0)=1.
6. Give an example of stationary process, which is not covariance stationary.

 

Section-B (5×8=40 marks)

Answer any FIVE questions. Each question carries EIGHT marks.

 

1. Show that a Markov chain is fully determined, when its initial distribution and the one step transition probabilities of the Markov chain are known.
2. Show that in a three dimensional symmetric random work, all the states are transient
3. Examine for the existence of stationary distribution {П_j } for the Markov chain, whose transition probability matrix is specified by p₀₁=1,

p_ij = q_i    if j=i-1

= p_i    if j=i+1, where i=1,2,3… and p_i+q_i=1

1. Under the condition X(0) =1, obtain the mean and variance of Yule process.
2. For a M│M│1 queueing system, show that the queue length process is Markov. Obtain the distribution of the waiting time in steady state.Also find E(N) and E(W_Q).
3. Define the term Renewal process. Derive the renewal function, if the inter renewal times have density

f(x)=λ²e^{– λx}x, x>0, λ>0

 

 

1. Let Y₀=0 and Y₁,Y₂,Y₃,… be independent and identically distributed random

variables with E(Y_k)=0 and E(Y²_k)=σ² , k=1,2,3,… Let X(0)=0 and

n

Xn =  (  Σ Y_k )²  – n σ² . Show that {X_n } is a Martingale with respect to {Y_n}

k=1

1. Show that the process {X(t)} defined by X(t) = Acos wt + Bsin wt, where A and B are uncorrelated random variables each with mean zero and variance unity, with w a positive constant, is covariance stationary.

 

Section-C (2×20=40)

Answer any TWO equestions. Each question carries TWENTY marks.

 

19.a) State and prove the necessary and sufficient condition for the state i of a

Markov chain to be recurrent.                                 (10 marks)

1. b) For a one dimensional symmetric random walk on the set of integers, find

f₀₀⁽ⁿ⁾ .                                                                           (10 marks)

20a) Show that states belonging to the same class have the same period.(8 marks)

1. b) Define stationary distribution { П _j} of a Markov chain. Prove that, for an

irreducible Markov chain with a doubly stochastic matrix, { П _j} is given by

П _j = 1/M, j=1,2,…,M,where M is the number of states.                                                                                                                    (4+8 marks)

21.a) State Chapman-Kolmogorov equation for a continuous time Markov chain.

Deduce Kolmogorov forward and backward equations.         (10 marks).

1. b) Obtain E[X(t)], where X(t) is a linear birth and death process.(10 marks).

22.a) Derive the integral equation satisfied by the renewal function of a Renewal

process.                                                                                           (8 marks)

1. b) If {X_n, n=0,l,2,…} is the Galton-Watson branching process, find E(X_n) and

Var(X_n).                                                                                           (12 marks)

 

 

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