LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

NO 20

FIFTH SEMESTER – APRIL 2008

ST 5400 – APPLIED STOCHASTIC PROCESSES

 

 

 

Date : 05/05/2008                Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION A

Answer ALL the questions

Each question carries 2 marks (10 X 2 =20 Marks)

 

1)   Explain continuous time Markov Chain.

2)   Give an example for discrete time Markov Chain.

3)   Define Poisson Process.

• Define recurrent state of a Markov Chain and give an example.
• What is renewal process?
• Define a pure birth process.
• Define a renewal counting process.

8)   Give an example of a Martingale.

9)   Give an example of a renewal process.

10) State renewal theorem.

SECTION B

Answer any 5 questions

Each question carries 8 marks (5  X 8 = 40 Marks)

 

• How do you relate the consumer’s brand switching behavior to a Stochastic model?
• Relate any queuing model to a discrete parameter space and discrete state space

stochastic model.

• Relate the time sharing computer system to a stochastic model.
• Explain one dimensional random walk.
• Suppose that the probability of a dry day (state 0) following a rainy day (state 1) is 1/3

and the probability of a rainy day following a dry day is  ½  . Here we have two state Markov chain.  (i) Find the transition  probability matrix (ii) Given that October 1 is a dry day what is the probability that October 3 is a dry day and October 5 is a dry day.

 

• A rat is put in to the maze as shown below. The rat moves through the compartments

at random. If there are k ways to leave the compartment he chooses each of these with probability 1/k. He makes one change of compartment at each instant of time. The state of the system is the number of compartment the rat is in. Determine the transition probability matrix.

 

 

17)  Explain the following processes with examples

1. Counting Process
2. Markov Process (Each carries 4 Marks)

 

 

 

18) Let Y₀ = 0, Y₁ , Y₂ ,Y₃ , …be independent rv’s with E( |Y_n | ) < , for all n and

E(Y_n ) = 0 , for all n. If X₀ = 0 and X_n = Show that { X_n } is a martingale with

respect to { Y_n }.

 

 

SECTION C

Answer any 2 questions

Each question carries 20 marks (2  X 20 = 40 Marks)

 

19) Let {X(t), t0} be a Poisson process. Find the distribution of X(t).

20 a )   Find the periodicity of the Markov Chain with the state space {0,1,2,3} and the

transition probability matrix

 

P =

 

20 b) Let {X_n ,n=0,1,2,3,…} be a sequence of iid rv’s with common probability

P(X₀ = i ) = p_i , i = 0, 1, 2,… Show that  {X_n ,n=0,1,2,3,…} is a Markov chain.

 

21 a) Let { X_t ,t e T) be a process with stationary independent increments when

T = {0,1,2,…}. Show that the process is a Markov process.

 

21 b) Consider the Markov chain with state space S={0,1,2,3,4} and one step Transition

probability matrix

 

 

Find the equivalence class and periodicity of states.

 

22) Explain the following in detail in the context of the appropriate applied scenario.

( Each Carries 5 Marks)

1. Stochastic Processes with discrete parameter and discrete state space.
2. Stochastic Processes with discrete parameter and continuous state space.
3. Stochastic Processes with continuous parameter and discrete state space.
4. Stochastic Processes with continuous parameter and continuous state space.

 

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