03.11.2003
1.00 - 4.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - STATISTICS
THIRD SEMESTER – NOVEMBER 2003
ST-3800/S915 - STOCHASTIC PROCESSES
Max:100 marks
SECTION-A
Answer ALL the questions.
(10x2=20 marks)
1. Define a stochastic process clearly explaining the time and state space.
2. Examine if a sequence of independent random variables possesses independent
increment property.
3. Define a Markov chain. Give an example.
4. Let the transition probability matrix of a Markov chain with the state space S= {0,1,2,3}
0 0 0 
 1
 0
1 0 0 

be P =
. Find the periodicities of the states.
1 / 3 2 / 3 0 0 


1 / 4 1 / 4 0 1 / 2
5. Define i) recurrence and ii) mean recurrence time of state i.
6. Describe a Poisson process.
7. Define current life and excess life associated with a renewal process.
8. Find the distribution of excess life if N(t) ~ P ( t).
9. If {Xn} is martingale with respect to {Yn} , Show that E [Xn+kY0 Y1 ….Yn] = Xn for all
k 0 .
10. Define Branching Process.
SECTION-B
Answer any FIVE questions.
(8x5=40 marks)
11. State and establish Chapman - Kolmogorov equations satisfied by a discrete time
Marhov - chain.
12. Show that d(i) = g cd {n  1  f ii(n ) > 0}.
13. Describe Yule process. Show that the marginal distirbution of the process is negative
binomial with p = e-t if the initial size is greater than 1.
14. Describe a Birth - Death process and derive kolmogorov forward differential equation.
15. Show that the renewal function corresponding to the life time density
2 x e-x , x > 0
f (x) =
0
,
t
elsewhere
1
(1  e  2 t ), t  0.
2
4
16. Define a renewal process. Show that renewal function M(t) satisfies the renewal
equation.
17. Let Y0 = 0, and Y1, Y2….be iid random variables with E (Yk) = 0 and
E (Y 2k ) = 2, k = 1, 2, ….
is given by
M (t) =
2

 n

Let X0 = 0 and Xn =   Yk   n  2 .
 k 1 
Show that {Xn} is a martingale w.r.t {Yn}.
18. Suppose the probability generating function of the off-spring distribution for a Branching
process is  (s) = 0.1 + 0.4 s + 0.5 S2. Obtain the extinction probability.
SECTION-C
Answer any TWO questions.
(2x20=40 marks)
19. i) Show that if i  j, then d(i) = d(j).
ii) Show that state i is recurrent if and only if
p
(n)
ii
.
(8)
(12)
n
20. i) Prove that the three dimensional symmetric random walk on the set of integers is a
transient Markov chain.
(15)
ii) Let {Xn, n  0} be an irreducible FMC with doubly stochastic tpm. Show that the
stationary probabilities are equal.
(5)
21. i) Show that the stationary distribution for a single server queueing model is geometric,
and also show that the distribution of waiting time is an exponential.
(7+8)
ii) Derive Kolnogorov forward differential equations for Telephone trunking model. (5)
22. i) Let the renewal counting process be Poisson. Find the joint distribution of  t and  t .
Deduce that the two random variables are independent.
(10)
ii) State and prove elementary renewal theorem.
(10)
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