LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2003

ST-5400/STA400 – APPLIED STOCHASTIC PROCESS

12.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION-A

Answer ALL questions.                                                                                   (10×2=20 marks)

 

1. Define a stochastic process with an example.
2. Define Bernoulli process with an example.
3. Give an example of a continuous time, discrete space stochastic process.
4. When do you say a process has stationary independent increments?
5. Define a Markov Chain.
6. Explain a doubly stochastic matrix.
7. Define the periodicity of a state i of a Markov Chain.
8. When the state i is said to be recurrent?
9. Explain the one dimensional random walk with an example.
10. Define a Martingale.

 

SECTION-B

 

Answer any FIVE  questions.                                                                          (5×8=40 marks)

 

11. Let {X_n, n=0, 1, 2, …} be a sequence of iid r.v’s with common distribution P(X_o = i) = p_I, iÎS, p_I>0, . Prove that {X_n, n = 0. 1,2, ….} is a Markov chain.
12. Prove that the state i of a Markov chain is recurrent iff .
13. Show that the state ‘0’ of a one-dimensional symmetric random walk is recurrent.
14. Explain a counting process with an example.
15. Explain a Poisson process with an example.
16. If {X₁ (t), t Î (o, ¥,} and {X₂ (t), t Î (o, ¥,} one two independent Poisson processes with parameters l₁ and l₂, Show that the distribution of X₁ (t) / (X₁ (t) + X₂(t) = n follows B

.

17. Explain in detail the generalization of a Poisson process.
18. Let {X_t, t Î T} be a process with stationary independent increments when T – 0,1,2….. show that the process is a Markov process.

 

 

 

 

 

SECTION-C

Answer any TWO  questions.                                                                          (2×20=40 marks)

 

19. a) Show that a communication is an equivalence relation.
20. b) Let{X_n, n ≥ 0} be a Markov Chain with state space S = {0,1,2,3,4} and transition

probability matrix      p=.     Find the equivalence classes.

20. Show that for a Poisson process with distribution of x(t) is given by P{x(t) = m} =

 

21. Explain the two dimensional symmetric random walk. Also prove that the state ‘0’ is recurrent.
22. a) Let Y_o­ = 0 andY₁, Y₂ .. be iid r.r’s with E(Y_k) = 0 and E(Y_k²) = s² ” k = 1,2, ….

Let X_n = .  Show that {X_n, n≥ 0 } is a martingale w.r.t {Y_n, n ≥ 0}.

1. b) Write short notes on Renewal process.

 

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