LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 42

THIRD SEMESTER – April 2009

ST 3809 – STOCHASTIC PROCESSES

 

 

 

Date & Time: 23/04/2009 / 1:00 – 4:00 Dept. No.                                                      Max. : 100 Marks

 

 

PART-A

             Answer all the questions:                                                                             (10 X 2 = 20)

1. 
   Define a stochastic process with an example.
2. Define a process with independent increments.

 

3. Show that communication between two states i and j satisfies transitive relation.

 

4. Define (i) transcient state (ii) recurrent state.

 

5. Define a Markov process.

 

6. Obtain the PGF of a Poisson process.

 

7. Define a renewal function. What is the relation between a renewal function and the

distribution functions of inter occurrence times?

 

8. When do you say that  is a martingale with respect to ?

 

9. What is a branching process?

 

10. What is the relationship between Poisson process and exponential distribution?

 

PART-B

            Answer any 5 questions:                                                                                    (5 X 8 = 40)

 

11. State and prove Chapman – Kolmogorov equation for a discrete time Markov chain.

 

12. Obtain the equation for in a Yule process with X(0) = 1.

 

13. Let and  be i.i.d random variables with mean 0 and variance.

Show that   is a martingale with respect to .

 

14. Show that the matrix of transition probabilities together with the initial distribution

completely specifies a Markov chain.

 

15. Show that the renewal function satisfies

 

 

16. Establish the relationship between Poisson process and Binomial distribution.

 

17. Obtain the stationary distribution for the Markov chain having transition probability

matrix

 

 

 

18. If a process has stationary independent increments and finite mean show that

 

where     and  .

 

PART-C

                 Answer any 2 questions:                                                                          (2 X 20 = 40)

 

19. a) State and prove the necessary and sufficient condition required by a state to be                               recurrent.

 

b.)  Verify whether state 0 is recurrent in a symmetric random walk in three dimensions.                                                                                                                                         (10+10)

 

20. a) State the postulates of a Poisson process. Obtain the expression for.

 

b.)  Obtain the distribution for waiting time of k arrivals for a Poisson process.                                                                                                                                                                    (15+5)

 

21. a) Obtain the generating function for a branching process. Hence obtain the mean and                        variance.

 

1.       b)  Let   be the probability that an individual in a generation generates k

off springs. If  obtain the probability of extinction.

(15+5)

22. a.) Obtain the renewal function corresponding to the lifetime density.

 

 

b.)  Show that the likelihood ratio forms a martingale.

 

c.)  Let be a martingale with respect to.  If  is a convex function with

 

show that   is a sub martingale with respect to .

                                                                                                                             (10+5+5)

                                        

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