17.04.2004
1.00 - 4.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
B.Sc., DEGREE EXAMINATION - STATISTICS
FIFTH SEMESTER – APRIL 2004
ST 5400/STA 400 - APPLIED STOCHASTIC PROCESSES
Max:100 marks
SECTION -A
(10  2 = 20 marks)
Answer ALL questions.
1.
2.
3.
4.
5.
6.
7.
Define a Stochastic Process.
What is 'State Space' of a Stochastic Process?
Define 'Counting Process'.
Explain 'Independent Increments'.
Define 'Markor Process'.
Define 'Transition Probability Matrix'.
Define 'accessibility' of a state from another.
2

 3 ? 0


1
P  0
?  is a stochastic matrix,
8. If
2


 ? ? 1


fill up the missing entries in the matrix.
9. Define 'Aperiodic' Markov chain.
10. Write down the postulates of 'Pure Birth Process'.
SECTION - B
Answer any FIVE questions.
(5  8 = 40 marks)
11. State the classifications of Stochastic Processes based on time and state space. Give an
example for each type.
12. Show that a sequence of independent random variables is a Markov Chain (M.C).
13. If P (X o  i) 1 / 3, i  0,1, 2, and the TPM is
3/4 1 / 4 0 
P  1/4 1 / 2 1 / 4 , find P (X2 = 2).
 0 3 / 4 1 / 4
14. Show that 'Communication' is an equivalence relation.
1
15. Classify the states of a M.C. whose TPM is
0
1
2
0 1 / 2 1 / 2 0
1 1 / 4 3 / 4 0
P 2 0
0
0

3 0
0 1/ 2
4  0
0
0
3 4
0 0 
0 0 
1 0 

0 1 / 2
1 0 
16. Describe a one-dimensional Random walk and write down its TPM.
17. State and prove any one property of a Poisson Process.
18. Write brief notes on: (a) Stochastic and Doubly Stochastic Matrices; (b) Extensions of
Poisson Process.
SECTION - C
(2  20 = 40 marks)
Answer any TWO questions.
19. a) Let { X(t) : t  T} be a process with stationary independent increments where
T = {0,1,2, ....}. Show that the process is a Markov Process.
n
b) If {Xn : n = 1,2,3, ...} is a sequence of i.i.d, r.v.s and Sn =  X i , n = 1,2,...., show that
i 1
{Sn} is an M.C.
(10+10)
20. a) Define 'recurrent' and 'transisiant' states. State (without proof) a necessary and
sufficient condition for a state to be recurrent.
b) Describe the two-dimensional random walk. Discuss the recurrence of the states.
(6+14)
21. State the posulates of a Poisson Process and derive the distribution of X(t).
22. a) Define a 'Martingale'.
If Yo = 0, Y1, Y2, ...., are i.i.d with E (Yn) = 0, V (Yn) = 2, show that:
n
b) Xn =  Yi is a Martingale with respect to {Yn}
i 1
 n 
c) Xn =   Yi 


 i 1 
2
- n 2 is a Martingale with respect to {Yn}.
(3+7+10)

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