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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI 600 034
M.Sc.DEGREE EXAMINATION STATISTICS
THIRDSEMESTER APRIL 2018
16PST3MC02 /ST3816/ST3812- STOCHASTIC PROCESSES
Date: 26-04-2018 Dept. No. Max. 100 Marks
Time: 09:00-12:00
Section A
Answer all the questions 10 X 2 20 marks
1. Define state space and time space of a stochastic process.
2. Define a transition probability matrix.
3. When a stochastic process is called Markov?
4. Write two properties of periodicity of a Markov chain.
5. Define recurrence and transient states of a Markov chain.
6. When a process is said to be covariance stationary
7. State the basic limit theorem of Markov process.
8. Define sub-martingale of a stochastic process.
9. Define a branching process.
10.State Abel lemma.
Section B
Answer any five questions 5 X 8 40 marks
11. Explain process with stationary independent increments Martingales
12. Define communication of states and show that communication is an equivalence relation.
13. State the necessary and sufficient condition for a state to be recurrent.
14. Derive a system of differential equations of a pure birth process.
15. Explain renewal process with two examples.
16. Show that the variance of a sum as a Martingale.
17. Establish the relationship of probability generating function for a branching process.
18. Explain stationary process with two examples.
Section- C
Answer any two questions 2 X 20 40 marks
19. (a)Show that the three dimensional random walk is transient.
Prove that state i is recurrent or transient according to whether Qii 1 or 0 respectively.
marks.
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20.(a) Derive backward and forward Kolmogorov differential equations for a birth and death process.
Derive the Yule process. (10 10) marks.
21.(a) Derive the mean and variance of branching process.
If is the probability of eventual extinction show that it satisfies the equation s .
(10 10) marks.
22.(a) Show that the moving average process is covariance stationary.
Explain Wald's martingale Doob's martingale. Right regular sequences and induced
Martingales for Markov chains. 14)marks.