LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
SUPPLEMENTARY EXAMINATION – JUNE 2007
ST 3809 / 3806 - STOCHASTIC PROCESSES
Date & Time: 27/06/2007 / 9:00 - 12:00
Dept. No.
Max. : 100 Marks
SECTION-A (10× 2=20 marks)
Answer ALL the questions
1. Define a discrete time Markov chain with stationary transition probabilities.
2. Let {Xn, n=0,1,2,…} be a Markov Chain with state space S={1,2,3} and transition
probability matrix
P=
1/2
2/3
3/5
1/4
0
2/5
1/4
1/3
0
Compute P[X3=3 | X1=2].
3. Write down the transition probability matrix for sums of the independent random
variables.
∞
4. Prove that for any state i and a transient state j, Σ pij(n)< ∞.
n=1
5. What do you understand by an absorbing barrier?
6. Define a Renewal process and its renewal function.
7. Write down the axioms of Brownian Motion Process.
8. Define a discrete time Branching Process.
9. Define Supermartingale.
10. What is a Markov Renewal Process?
SECTION-B (5×8=40)
Answer any FIVE questions
11. Prove that the one dimensional symmetric random walk on the set of integers is
an irreducible recurrent Markov chain.
12. Write down Chapman-Kolmogorov equations for a continuous time Markov
chain. Give an example of a non-Markov chain, whose transition probabilities
satisfy Chapman-Kolmogorov equations.
13. Define compound Poisson process. For this process, obtain the generating
function, expectation and variance.
14. Obtain the system of differential equations satisfied by the transition probabilities
of Yule process and calculate its transition probabilities when the initial condition
is X(0) =1.
15. What are the postulates governing the Birth and Death process? Obtain the
differential-difference equations governing this process.
16. Define M/M/1 queue. Obtain E(WQ) in this case, when steady state solution
exists.
17. Let {Yn} be a sequence of independent unit normal random variables and
2
Sn= (Y1+Y2+…+Yn). Let Xn=1/(√n+1) eSn /2(n+1). Show that {Xn} is a
Martingale with respect to {Yn}.
18. Show that a stationary process with finite second moments is covariance
stationary. Show that a covariance stationary process is not necessarily a
stationary process.
SECTION-C (2×20=40 marks)
Answer any TWO questions
19.a. Obtain the necessary and sufficient condition for the state i of a Markov chain
to be recurrent.
b. State and prove Chapman-Kolmogorov equations for a discrete time Markov
chain.
c. Show that the states belonging to the same class have the same period.
(10+6+4 marks)
20.a. State and prove the Basic limit theorem of Markov chains(12 marks)
b. Prove that in a finite irreducible Markov chain with s states, any state can be
reached from any other state in s steps or less (8 marks)
21.a. If {Xn, n=0,1,2,…}is the Galton-Watston,obtain E(Xn) and Var(Xn)
(12 marks)
b. Define the extinction probabilities associated with the Branching process
(Xn, n=0,1,2,…} and prove in the usual notation that the probability of
extinction is the smallest positive root of the equation φ(s)= s, where φ (s) is the
generating function of the off spring distribution. (8 marks)
22.a. Find the mean and variance of the Renewal Process {N(t), t≥0}.
b. Derive the integral equation satisfied by the renewal function of a Renewal
process.
****************