LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

YB 20

FIFTH SEMESTER – April 2009

ST 5400 – APPLIED STOCHASTIC PROCESSES

 

 

 

Date & Time: 28/04/2009 / 1:00 – 4:00  Dept. No.                                                   Max. : 100 Marks

 

 

SECTION – A

Answer all the questions                                                                                             (10 x 2 = 20 )      

               

1. Define Stochastic Process with an example.
2. What is the State Space of a Stochastic Process?
3. Define Markov Process.
4. Explain Independent Increments.
5. Define Transition Probability Matrix.
6. If P = is a Stochastic Matrix, fill up the missing entries in the Matrix.
7. Define Accessibility of a State from another state.
8. What is a Recurrent State?
9. Define aperiodic Markov Chain.
10. What is a Martingale?

SECTION – B

Answer any Five questions                                                                                          (5 x 8 = 40)    

  

11. State the classification of Stochastic Processes based on time and state Space. Give an example for each type.
12. Describe One-dimensional Random Walk and write down its Transition Probability Matrix.

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

probability matrix

and the initial distribution 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},     iv)   Pr{X₂ = 2, X₁ = 1, X₀ = 2}

14. Show that Communication is an equivalence relation.
15. Consider the following Transition Probability Matrix.  Using the necessary and sufficient condition for recurrence, examine the nature of all the three states.
16. State any one property of Poisson Process.

 

 

 

 

17. Classify the states of Markov Chain with Transition Probability Matrix

 

18. State and stablish Chapman – Kolmogorov equations for a discrete time Markov chain.

 

SECTION – C

Answer any Two questions                                                                                       (2 x 20 = 40)        

              

19. Derive the distribution of X(t), is a Poisson Process.                               (20)
20. a). State and prove a necessary and sufficient condition for a state to be Recurrent .

b). Explain the Two-dimensional Symmetric Random Walk.                                   (10+10)

21. (a) Let { Z_i, i = 1,2…} be a sequence of independent identically distributed random variables   with mean 1.  Show that   X_n =   is a Martingale                                             (8)

(b). Consider a Markov Chain with Transition Probability Matrix

 

Find the equivalence classes and compute the periodicities of all the four states.           (12)

 

22. a). Illustrate with an example Basic Limit Theorem of Markov chains..

b). Consider the following Transition Probability Matrix explaining seasonal changes on

successive day. (S – Sunny, C – Cloudy)

Today state

 

yesterday state

Compute the stationary probabilities and interpret the results.                            [10+10]

 

 

Go To Main page

Latest Govt Job & Exam Updates:

View Full List ...