LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AB 09

FIFTH SEMESTER – NOV 2006

ST 5400 – APPLIED STOCHASTIC PROCESSES

(Also equivalent to STA 400)

 

 

Date & Time : 03-11-2006/9.00-12.00     Dept. No.                                                   Max. : 100 Marks

 

 

 

SECTION A

ANSWER ALL QUESTIONS.                                                                               (10 X 2 =20)

 

1. Give an example of discrete time –continous state stochastic process  ?
2. Give one example each for “communicative states” and “non communicative states”
3. Define : Markov process
4. When do you say a stochastic process has “Stationary Independent Increments” ?
5. Identify the closed sets corresponding to a Markov chain with transition probability matrix.
6. What is a recurrent state ?
7. When do you say a given state is “aperiodic” ?.
8. What is a doubly stochastic matrix.?
9. Name the distribution associated with waiting times in Poisson process
10. What is a martingale ? .

 

SECTION B

Answer any FIVE questions                                                                      (5 X8 =40)

 

1. Show that a one step transition probability matrix of a Markov chain is  a stochastic matrix.

 

1. Write a detailed note on classification of stochastic processes

 

1. Show that every stochastic process with independent increments is a Markov process.

 

1. Obtain the equivalence classes corresponding to the  Transition Probability Matrix

 

.

 

1. Consider the following Transition Probability Matrix . Using a necessary and sufficient condition for recurrence, examine the nature of all the three states.

 

1. Form the differential equation corresponding to Poisson process

 

 

1. Messages arrive at a telegraph office in accordance with the laws of a Poisson process with mean rate of 3 messages per hour. (a) What is the probability that no message will have arrived during the morning hours (8,12) ? (b) What is the distribution of the time at which the fist afternoon message arrives ?

 

1. Show that, under usual notations,

 

 

SECTION C

Answer TWO questions.                                                                            (2 X 20 =40)

 

1. (a) Let be a sequence of random variables with mean 1.Show thatis a Martingale.            (8)

(b) Consider a Markov chain with TPM . Find the equivalence classes and compute the periodicities of all the 4 states  (12)

 

1. (a) Illustrate with an example how Basic limit theorem can be used to relate stationary distributions and mean time of first time return.                      (8)

(b) Suppose that the weather on any day depends on the weather conditions for the previous two days. To be exact, suppose that if it was sunny today and yesterday, then it will be sunny tomorrow with probability 0.8; if it was sunny today but cloudy yesterday then it will be sunny tomorrow with probability 0.6; if it was cloudy today but sunny yesterday, then it will be sunny tomorrow with probability 0.4; if it was cloudy for the last two days, then it will be sunny tomorrow with probability 0.1. Transform the above model into a Markov chain and write down the TPM. Find the stationary distribution of the Markov chain. On what fraction of days in the long run is it sunny ?  (12)

 

1. Derive under Pure-Birth Process assuming
2. Write short notes on any of the following :

(a) One dimensional random walk      (5)

(b) Periodic states       (5)

(c) Martingales            (5)

(d) Properties of Poisson Process        (5)

 

Go To Main page

Latest Govt Job & Exam Updates:

View Full List ...