LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2012

ST 3812/3809 – STOCHASTIC PROCESSES

 

 

Date : 24-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Section – A

       Answer all the questions:                                                                                         10 x 2 = 20 marks

1. Define convergence in quadratic mean.
2. Define periodicity and aperiodicity of a Markov chain.
3. Give an example for a reducible Markov chain.
4. Write the infinitesimal generator of a birth and death process.
5. Write any two applications of Poisson process.
6. Provide any two examples for renewal process.
7. Define a super martingale.
8. Define discrete time branching process.
9. Write a note on stationary process.
10. Write different types of stochastic processes.

                                                                                 

Section – B

Answer any five questions:                                                                                          5 x 8 = 40 marks

11. Explain (i) martingale   (ii) point process
12. Explain one-dimensional random walk.
13. (a) Show that a state i is recurrent if and only if _iiⁿ = .

• If i j and if i is recurrent show that j is recurrent.                                              (4+4)

14. Derive P_n(t) for the Yule process with X(0) = 1.
15. Derive the mean for a birth and death process if λ_n = nλ + a and μ_n = nμ with λ > 0 , μ >0 and a>0.
16. Explain (i) renewal function   (ii) excess life   (iii) current life  (iv) mean total life
17. Explain Markov branching process with three examples.
18. Write a note about (i) stationary process on the circle (ii) stationary Markov chains.

                                                                                  

 

Section – C

Answer any two questions:                                                                                        2 x 20 = 40 marks

 

19. (a) Show that state 0 is recurrent for a two dimensional random walk.

(b) Derive the basic limit theorem of Markov chains.                                                   (5+15)

 

20. For the gambler’s  ruin  on (n+1) states  with  P(X_n+1 = i+1  |  X_n = i ) = p ,

P(X_n+1 = i-1  |  X_n = i ) = q  and  0 and n are absorbing states , calculate u_i  = _i  (C₀ )  and  v _i ( C_n) .

21. (a) Derive the differential equations for pure birth process.

(b) Derive P_n(t)  for Yule process  with X(0) = N.                                                      (10 + 10)

22. (a) State and prove the basic renewal  theorem.

(b) Derive mean and variance of  branching process.                                                (10 + 10)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2012

ST 3812 – STOCHASTIC PROCESSES

 

 

Date : 03/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer all the questions:                                                                                          (10 x 2 = 20 Marks)

 

1. Define a point process.
2. Define n step transition probability.
3. Write any two basic properties of the period of a state.
4. If i « j and if i is recurrent then show that j is also recurrent.
5. Define mean recurrence time.
6. What is the infinitesimal generator of a birth and death process?
7. Define excess life and current life of a renewal process.
8. Define a sub martingale.
9. Write down the postulates of a birth and death process.
10. Write down any two examples for stationary process.

 

 

 

SECTION – B

Answer any Five questions:                                                                                     (5 x 8 = 40 Marks)

 

11. Explain (i) process with stationary independent increments (ii) Markov processes.
12. Explain spatially homogenous Markov chains.
13. Prove that a state i is recurrent if an only if

å  P_iiⁿ  =  ∞

14. For a two dimensional random walk, prove that å  P₀₀ⁿ  =  ∞
15. Determine stationary probability distribution for a random walk whose transition probability matrix is

 

0          1          0          0     . . .

q₁         0          p₁         0     . . .

0          q₂         0          p₂   . . .

P  =      .

.

.

 

 

 

 

 

-2-

 

16. Derive P_n (t) for a Poisson process.
17. Derive the expected value of a birth and death process with linear growth and immigration.
18. State and prove the basic renewal theorem.

 

 

 

SECTION – C

Answer any two questions:                                                                                      (2 x 20 = 40 Marks)

 

19. (a) State and prove the basic limit theorem of Markov chains.

(b)  Explain discrete renewal equation.                                                           (15 + 5)

 

20. (a) Derive the differential equations for a pure birth process.

(b)  Derive the Kolmogorov forward and backward differential equations of a birth and

death process.                                                                                          (10 +10)

 

21. (a) Explain renewal function, excess life, current life and mean total life.

(b)  If {X_t}is a renewal process with μ = E [X_t] < ∞ , then show that

lim 1/t M (t) = 1/μ  as t ® ∞                                                                   (8 + 12)

22. (a) Show that π is the smallest positive root of the equation j(s) = s

for a branching process.

(b)  Compute expectation and variance of branching process.                       (10 + 10)

 

 

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