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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 25

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – November 2008

    ST 3809 / 3800 – STOCHASTIC PROCESSES

 

 

 

Date : 05-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART-A

Answer all questions:                                                                                      (10×2=20)

 

1. Define a Markov process.

 

2. Define recurrent state and transient state of a Markov chain.

 

3. Define a Martingale of the process {Xn} with respect to {Yn}.

 

4. Obtain E[X₁ + X₂ +…..+ X_N] where Xi, i=1, 2, 3,….. are i.i.d and independent of the

random variable N .

 

5. Let X₁, X₂ be independent exponentially distributed random variables parameters λ₁

and λ₂ respectively. Obtain P[min(X₁,X₂)>t] .

 

6. Messages arrive at the telegraph office in accordance with the laws of a Poisson

Process with mean rate of 3 messages per hour. What is the probability of getting no

message during morning hours from10 to 12?

 

7. Obtain the pgf of a Poisson process.

 

8. If X₁ and X₂ are independent random variables with distribution functions of F₁ and F₂ respectively. Write

an expression for the distribution function of X=X₁+X₂?

 

9. Obtain P[N(t)=k] in terms of the distribution functions of the life times for a renewal

Process?

 

10. Define a stationary process.

 

PART-B

Answer 5 questions:                                                                                        (5×8=40)

 

11) Consider the Markov chain with states 0,1,2 having the TPM

 

 

and  P[X₀ = i] = 1/3,  i = 1,2,3

Obtain i) P[X₂=0]

1. ii) P[X₂=0, X₁=2/ X₀=1]

iii) P[X₂=0, X₁=2, X_0­=1]                                                              (4+2+2)

 

12) Verify whether the Markov chain with TPM given below is ergodic

 

 

 

 

 

13) Show that for a renewal process in the usual notation,

M(t)= F(t) + F*M(t)

 

14) Prove that if {Xn} is a super martingale with respect to {Yn} then

1. i) E[Xn+k ç Y₀,Y₁,…..Yn ] ≤ Xn,
2. ii) E[Xn­] ≤ E[Xk], 0 ≤ k ≤ n

 

15) State the postulates of birth and death process. Obtain the forward differential

equations for a birth and death process.

 

16)  Obtain the Stationary distribution of a Markov chain with TPM

 

17) Consider the times {S_k} at which the changes of Poisson process X(t) occur. If

Si = T₀ + T₁ + … + T_i-1, i = 1,2,3,… obtain the joint distribution function of S₁,

S₂,……S_n given X(t) = n.

 

18) Show the periodicity is a class property.

 

PART-C

Answer 2 questions:                                                                                        (2 x 20=40)

 

19) a) Show that i is recurrent if and only if ∑P_ii n = ∞

1. b) Show that in a one dimensional symmetric random walk state 0 is recurrent.
2. c) if j is transient prove that for all i ∑P_ij n < ∞                                              (8+7+5)

 

20) a) State the postulates of a Poisson process and obtain the expression for P_n(t).

 

1. b) If X(t) has a Poisson process, u<t, k<n obtain P[X(u) = k çX(t) = n]             (12+8)

 

21) a) Obtain the renewal function corresponding to the lifetime density

f(x) = λ² x e ^{– λ x} ,  x ≥ 0

1. b) Let Y₀=0, Y₁, Y₂,….. be i.i.d with

E[Y_k] = 0    var[Y_k]=σ²     k=1,2,……

E[| Y_n |] < ∞   let X₀=0

Show that

1. i) X _n= Y_i
2. ii) X_n = (Y_i )² – nσ²

are martingales.                                                                                                 (10+5+5)

 

22) a) Derive the p.g.f of a branching process. Hence obtain the mean and variance of X_n.

 

1. b) Let the offspring distribution be P[ζ= i] = 1/3 , i = 0,1,2

Obtain the probability of extinction.

 

 

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