Loyola College B.Sc. Statistics April 2012 Applied Stochastic Processes Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – APRIL 2012

ST 5400 – APPLIED STOCHASTIC PROCESSES

 

 

Date : 27-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

 

Section-A                                                     10×2=20 marks

Answer all the  questions.

  • Define the term Stochastic
  • Define State space with an example
  • What is meant by Martingales?
  • Define the period of a state of a Markov Chain.
  • When is state “I” communicate with state “j”?
  • What is meant by transient state?
  • What is meant by Stationary Increments?
  • Define irreducible Markov Chain with an example
  • Explain the term transitivity.
  • Define mean recurrence time.

 

Section-B                                                                                  5×8=40 marks

Answer any FIVE  questions.

11) Discuss the  applications of   Stochastic processes  with suitable illustrations..

12) Explain the Gambler’s ruin  problem with the TPM .

13) Explain the one dimensional random walk problem with the TPM

14) If ‘’I” communicate with “j” and “I” is recurrent then show that “j” is also recurrent.

15) Discuss in detail the higher order transition probabilities with suitable illustration.

 

 

 

 

16) Find the Stationary distribution of a Markov Chain with States 1,2 and 3 with the following

TPM

17)  Show that recurrence is a class property.

18) Explain two dimensional random walk..

Section-C                                                                                  2×20=40 marks

Answer any  TWO  questions.

19a)  If the  probability of a dry day (state-0) following a rainy day (state-1)is 1/3, and that of a rainy day following a dry day is  ½.   Find

  1. Probability that May 3 is a dry day given that May first is a dry day.
  2. ii) Probability that May 5 is a rainy day given that May first is a dry day.

19b)  Discuss in detail Pure Birth process.   (12 + 8 Marks)

20a) State and prove Chapman-Kolmogrov equation.

20b) Discuss in detail the applications of basic limit theorem of Markov Chains. (12 + 8 Marks)

21) A white rat is put into the maze consisting of 9 compartments. The rat moves through the compartment at random. That is there are k ways to leave a compartment. The rat chooses each of the move with probability1/k.

  1. a) Construct the Maze

b)The Transition probability matrix

  1. c) The equivalence class
  2. d) The periodicity (5+5+5+5 Marks)

 

22) Diabetes disease in any  Society (with different classes of people ) often considered as a family disease which occurs as  successive generations in a family can be regarded as a Markov Chain. Thus the disease of the children is assumed to be depended only on the disease of the parents. The TPM of such model is as follows:

Children’s Class

Mild    Moderate   Severe

Mild                      0.40.        0.50         0.10

Parent’s Class    Moderate                0.05         0.70          0.25

Severe                     0.05         0.50          0.4

Find a) What proportion of   people are Moderate class in the long run suffering from diabetes?

  1. b) Show that the MC is recurrent  (12 + 8 Marks)

 

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