LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
FIFTH SEMESTER – NOVEMBER 2012
ST 5400 – APPLIED STOCHASTIC PROCESSES
Date : 10/11/2012 Dept. No. Max. : 100 Marks
Time : 9:00 – 12:00
Section-A
Answer all the questions: (10×2=20 marks)
- Give an example for one and two dimensional Stochastic Processes.
- Define Time space with an example.
- Define Null recurrence.
- What is meant by Periodicity?
- Briefly explain the term random walk.
- Define the term communication of the states.
- What is meant by absorbing state?
- What is meant by TPM?.
- Define Markov Chain.
- What is meant by Birth process
Section-B
Answer any FIVE questions: ( 5×8=40 marks)
11)Discuss in detail the classifications of the Stochastic Processes.
12) Distinguish between Symmetry and Transitivity of communication with an example.
13) Discuss in detail any two applications of Stochastic modeling. .
14) Explain the Gambler’s Ruin problem with an example.
15)Discuss the applications of stationary distribution with suitable illustration.
16) Discuss in detail the higher order transition probabilities with suitable illustration.
17) 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.
- a) Construct the Maze
b)The Transition probability matrix
18) Discuss the Social Mobility problem.
Section-C
Answer any TWO questions: ( 2×20=40 marks)
19a) Show that a Markov Chain is fully determined, when its initial distribution and one step transition
probabilities of the Markov chain are known.
19b) State and prove Chapman-Kolmogrov equation.
20) Sociologist often assumes that the social classes of a successive generation in a family can be regarded as a Markov chain. The TPM of such model is as follows.
| Son’s Class | ||||
| Lower | Middle | Upper | ||
| Lower | 0.4 | 0.5 | 0.1 | |
| Father’s Class | Middle | 0.05 | 0.7 | 0.25 |
| Upper | 0.05 | 0.5 | 0.45 | |
Find
- What proportion of people are lower class in the long run?
- What proportion of people are middle class in the long run?
- What proportion of people are upper class in the long run?
21a) Explain the one dimensional random walk problem with the TPM .
21b) 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 i) Probability that May 3 is a dry day given that May first is a dry day. ii) Probability that May 5 is a rainy day given that May first is a dry day..
22) Write short notes on the following
- a) Poisson Process
- b) Irreducible Markov Chain
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