LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 

BA 14

FIFTH SEMESTER – November 2008

ST 5400 – APPLIED STOCHASTIC PROCESSES

 

 

 

Date : 12-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

 

Answer ALL the questions:                                                                (10 x 2 = 20)

                  

1. What is meant by a stochastic process?

1. What is a state space of a stochastic process?
2. Explain ‘Independent Increments’.
3. Define ‘Markov process’.
4. Define Transition Probability Matrix.

1. Define accessibility of a state from another state.

7. Ifis a stochastic matrix, fill up the missing entries in the

matrix.

8. Define aperiodic Markov chain.
9. Define absorbing state.
10. Define irreducible Markov chain.

                                                                                      

Section – B

Answer any FIVE of the following:                                                                       (5 x 8 = 40)

                                               

11. State the classification of stochastic processes based on time and state space.

Give an example for each type.

12. Prove that a Markov chain is completely determined by the one step transition  

       probability matrix and the initial distribution.

13. Let { X_n, n ³ 0} be a Markov chain with three states 0,1 and 2. If the transition

       probability matrix is

                                  

                                   

 

and the initial distribution is Pr{ X₀ = i}= 1/3, i = 0,1,2,

            find

            i).    Pr{X₁ = 1 ½ X₀ = 2}

            ii).   Pr{X₂ = 2 ½ X₁ = 1}

            iii).  Pr{X₂ = 2, X₁ = 1 ½ X₀ = 2} and

            iv).  Pr{X₂ = 2, X₁ = 1, X₀ = 2}

 

 

 

 

14. Obtain the equivalence classes corresponding to the transition probability matrix

 

 

15. Form the differential – difference equation corresponding to Pure birth process.
16. Describe the one dimensional random walk and write down its tpm.
17. Describe second order process, covariance function and its properties.
18. Derive any one property of Poisson process.

      

                                                          Section – c

 

Answer any TWO of the following:                                                             (2 X 20 =40)

 

 19.a) Let { Z_i, i = 1,2…} be a sequence of random variable with mean 0. Show that

       

           X_n =   is a Martingale.

1. b) Consider a Markov chain with state space {0, 1, 2, 3} and tpm 

 

 

P  = 

            

            

 

 

   Find the equivalence classes and compute the periodicities of all the 4 states

      

20. Sociologists often assume that the social classes of successive generations in

              family can be regarded as a Markov chain. Thus the occupation of son is

              assumed to depend only on his father’s occupation and not his grandfather’s.

              suppose that such a model is appropriate and that the transition probability

              matrix is given by

                                               

 

               

 

               For such a population what fraction of people are middle class in the long run?

21. Define the Poisson process and find the expression for P_n(t).
22. Write short notes on the following:

(a). Stationary distribution

(b). Communicative sets and their equivalence property

(c). Periodic states

 

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