LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2006

                                                   ST 3806 – STOCHASTIC PROCESSES

 

 

Date & Time : 17-04-2006/AFTERNOON   Dept. No.                                                       Max. : 100 Marks

 

 

SECTION  A                           Answer all the questions                           10 ´ 2 = 20

 

1. Define stationary independent increment process.
2. Show that the square of a stochastic matrix is stochastic.
3. Suppose the one-step tpm is an identitiy matrix, show that the states are all recurrent.
4. Find a stationary disribution of an MC with one-step tpm

 

P =       0.3    0.7

0.7    0.3

5. For a Poisson process, find the covariance function.
6. Describe Pure birth process.
7. Define excess life and current life of a Renewal process.
8. For a martingale { X_n , n = 0.1.2,…}, show that E(X_n) = E(X_{n +1}), n = 0,1,2,…
9. Describe a branching process.
10. Define a covariance stationary process and give an example.

 

SECTION  B                               Answer any five questions                       5 ´ 8 = 40

 

11. For a stationary independent increment process, show that the variance of the marginal distribution is linear in the time parameter.
12. Define periodicity and show that it is a class property.
13. If the one step tpm of an irreducible finite state Markov Chain is symmetric, show that the stationary distribution is uniform.
14. Describe a Poisson Process and derive its marginal distribution.
15. For a linear growth process with immigration, find the average size of the population if the initial population is i units.
16. Derive the generating function relations satisfied by a Branching process.
17. If { X(t), t ³ 0} is a Brownian motion process, show that the distribution of (X(t₁),X(t₂)) is bivariate normal.
18. If the interoccurrence distribution of a Renewal process is exponential, find the distributions of i ) current life and ii ) excess life.

 

 

 

 

 

 

SECTION  C                              Answer any two questions                      2 ´ 20 = 40

 

19 a). State and establish Chapman-Kolmogorov equations satisfied by a Markov Chain.

b).   Illustrate Basic limit theorem with an example.

20 a). Describe Birth-Death process. Derive Kolmogorov backward equations satisfied by the Birth-Death process.

1. b) Describe telephone trunking model and find its stationary distribution.

 

21 .a) State and prove Elementary renewal theorem in Renewal theory.

1. b) Find the renewal function associated with a renewal process having the interoccurrence distribution with pdf

 

f(x) = l² x exp(-l x), x > 0, l > 0.

 

22 a) Let { X_n , n = 0.1.2,…}be a Branching process with X_{0 =} 1. Find the mean and variance of  X_n in terms of those of the offspring distribution.

1. b) Let { X_n , n = 0.1.2,…}be a covariance stationary process with zero mean and the covariance function R_x(v) . Find the best predictor of X_n+1 of the form aX_n, where a is a real constant.

 

 

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