LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

  FIFTH SEMESTER – APRIL 2004

ST 5400/STA 400 – APPLIED STOCHASTIC PROCESSES

17.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION -A

 

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

 

1. Define a Stochastic Process.
2. What is ‘State Space’ of a Stochastic Process?
3. Define ‘Counting Process’.
4. Explain ‘Independent Increments’.
5. Define ‘Markor Process’.
6. Define ‘Transition Probability Matrix’.
7. Define ‘accessibility’ of a state from another.
8. If  is a stochastic matrix,

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 and the TPM is

,  find P (X₂ = 2).

 

14. Show that ‘Communication’ is an equivalence relation.

 

 

 

 

 

 

 

15. Classify the states of a M.C. whose TPM is

 

0        1        2     3      4

 

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

 

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

 

19. a) Let { X(t) : t Î T} be a process with stationary independent increments where

T =  {0,1,2, ….}.   Show that the process is a Markov Process.

1. b) If {X_n : n = 1,2,3, …} is a sequence of i.i.d, r.v.s and S_n = n = 1,2,…., show that

{S_n} 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.

1. 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 Y_o = 0, Y₁, Y₂, …., are i.i.d with E (Y_n) = 0, V (Y_n) = s², show that:

1. b) X_n = is a Martingale with respect to {Y_n}
2. c) X_n = – n s² is a Martingale with respect to {Y_n}.                               (3+7+10)

 

 

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