LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

  FIFTH SEMESTER – APRIL 2004

ST 5400/STA 400 – APPLIED STOCHASTIC PROCESSES

17.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION -A

 

Answer ALL questions.                                                                              (10 ´ 2 = 20 marks)

 

1. Define a Stochastic Process.
2. What is ‘State Space’ of a Stochastic Process?
3. Define ‘Counting Process’.
4. Explain ‘Independent Increments’.
5. Define ‘Markor Process’.
6. Define ‘Transition Probability Matrix’.
7. Define ‘accessibility’ of a state from another.
8. If  is a stochastic matrix,

fill up the missing entries in the matrix.

9. Define ‘Aperiodic’ Markov chain.
10. Write down the postulates of ‘Pure Birth Process’.

 

SECTION – B

 

Answer any FIVE questions.                                                                                     (5 ´ 8 = 40 marks)

 

11. State the classifications of Stochastic Processes based on time and state space. Give an example for each type.
12. Show that a sequence of independent random variables is a Markov Chain (M.C).
13. If and the TPM is

,  find P (X₂ = 2).

 

14. Show that ‘Communication’ is an equivalence relation.

 

 

 

 

 

 

 

15. Classify the states of a M.C. whose TPM is

 

0        1        2     3      4

 

16. Describe a one-dimensional Random walk and write down its TPM.

 

17. State and prove any one property of a Poisson Process.

 

18. Write brief notes on: (a) Stochastic and Doubly Stochastic Matrices; (b) Extensions of Poisson Process.

 

SECTION – C

 

Answer any TWO questions.                                                                       (2 ´ 20 = 40 marks)

 

19. a) Let { X(t) : t Î T} be a process with stationary independent increments where

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

1. b) If {X_n : n = 1,2,3, …} is a sequence of i.i.d, r.v.s and S_n = n = 1,2,…., show that

{S_n} is an M.C.                                                                                                  (10+10)

 

20. a) Define ‘recurrent’ and ‘transisiant’ states. State (without proof) a necessary and

sufficient condition for a state to be recurrent.

1. b) Describe the two-dimensional random walk. Discuss the recurrence of the states.

(6+14)

21. State the posulates of a Poisson Process and derive the distribution of X(t).

 

22. a) Define a ‘Martingale’.

If Y_o = 0, Y₁, Y₂, …., are i.i.d with E (Y_n) = 0, V (Y_n) = s², show that:

1. b) X_n = is a Martingale with respect to {Y_n}
2. c) X_n = – n s² is a Martingale with respect to {Y_n}.                               (3+7+10)

 

 

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                         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)

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 20

B.Sc.  DEGREE EXAMINATION –STATISTICS

FIFTH SEMESTER – APRIL 2007

ST 5400 – APPLIED STOCHASTIC PROCESSES

 

 

Date & Time: 03/05/2007 / 1:00 – 4:00          Dept. No.                                                     Max. : 100 Marks

 

 

SECTION A

ANSWER ALL QUESTIONS.                                                                               (10 X 2 =20)

 

1. Give an example for a discrete time and continuous state space stochastic process
2. Define : Covariance Stationary.
3. Give an example of an irreducible Markov chain.
4. Give an example for a  stochastic process having “Stationary Independent Increments” ?
5. Mention the usefulness of classifying a set of states as closed or not in a Markov chain.
6.  Define the term : Mean recurrence time.
7. When do you say a given state is “positive recurrent” ?.
8. Give an example for symmetric random walk…
9. Under what condition pure birth process reduces to Poisson process ?
10. For what values of p and q the following transition probability matrix becomes a doubly stochastic matrix ?

SECTION B

Answer any FIVE questions                                                                      (5 X8 =40)

 

1. Consider the process where and are uncorrelated random variables with mean 0 and variance 1. Find mean and varaince functions and examine whether the process is covariance stationary

 

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

 

1. Show that a Markov chain is completely determined if its transition probability matrix and the distribution of is known.

 

1. A player chooses a number from the set of all non negative integers. He is paid an amount equivalent to the number he gets. Write the TPM corresponding to his earnings, given that probability of getting the number is
2. Consider the following Transition Probability Matrix explaining seasonal changes on successive days (S- Sunny, C-Cloudy)

Today

 

(S,S)  (S,C)      (C,S) (C,C)r

 

 

(S,S)    0.8       0.8       0          0

 

Yesterday        (S,C)    0          0          0.4       0.6

 

(C,S)    0.6       0.4       0          0

 

(C,C)   0          0          0.1       0.9

 

Compute the stationary probabilities and interpret your results

 

 

 

 

1. A radioactive source emits particles at a rate of 5 per minute in accordance with a poisson process. Each particle emitted has a probability of 0.6 of being recorded, Find in a 4 minute interval the probability that the number of particles recorded is 10..

 

1. Given the transition probability matrix corresponding to the Markov chain with states {1,2,3,4}Find the probability distribution of n which stands for the number of steps needed to reach state 2 starting from the same and also find its mean. Offer your comments regarding the state 2

 

 

1. Obtain the differential equation corresponding to Poisson process

 

 

SECTION C

Answer TWO questions.                                                                            (2 X 20 =40)

1. (a) Let be a sequence of random variables with mean 0 and is a Martingale.    (10)

(b) Show that every stochastic process with independent increments is a Markov process.

 

1. (a)  Show that for a process with independent increments is linear in t                                                                                                 (12)

(b) State and prove the reproductive property of Poisson process     (8)

1. Explain the postulates of Yule-Furry process and find an expression for
2. Write short notes on the following

(a) Stationary Distribution      (5)

(b) Communicative sets and their equivalence property (5)

(c) Interarrival time in Poisson process (5)

(d) Periodic States (5)

 

 

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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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

 

BA 14

FIFTH SEMESTER – November 2008

ST 5400 – APPLIED STOCHASTIC PROCESSES

 

 

 

Date : 12-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

 

Answer ALL the questions:                                                                (10 x 2 = 20)

                  

1. What is meant by a stochastic process?

1. What is a state space of a stochastic process?
2. Explain ‘Independent Increments’.
3. Define ‘Markov process’.
4. Define Transition Probability Matrix.

1. Define accessibility of a state from another state.

7. Ifis a stochastic matrix, fill up the missing entries in the

matrix.

8. Define aperiodic Markov chain.
9. Define absorbing state.
10. Define irreducible Markov chain.

                                                                                      

Section – B

Answer any FIVE of the following:                                                                       (5 x 8 = 40)

                                               

11. State the classification of stochastic processes based on time and state space.

Give an example for each type.

12. Prove that a Markov chain is completely determined by the one step transition  

       probability matrix and the initial distribution.

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

       probability matrix is

                                  

                                   

 

and the initial distribution is 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} and

            iv).  Pr{X₂ = 2, X₁ = 1, X₀ = 2}

 

 

 

 

14. Obtain the equivalence classes corresponding to the transition probability matrix

 

 

15. Form the differential – difference equation corresponding to Pure birth process.
16. Describe the one dimensional random walk and write down its tpm.
17. Describe second order process, covariance function and its properties.
18. Derive any one property of Poisson process.

      

                                                          Section – c

 

Answer any TWO of the following:                                                             (2 X 20 =40)

 

 19.a) Let { Z_i, i = 1,2…} be a sequence of random variable with mean 0. Show that

       

           X_n =   is a Martingale.

1. b) Consider a Markov chain with state space {0, 1, 2, 3} and tpm 

 

 

P  = 

            

            

 

 

   Find the equivalence classes and compute the periodicities of all the 4 states

      

20. Sociologists often assume that the social classes of successive generations in

              family can be regarded as a Markov chain. Thus the occupation of son is

              assumed to depend only on his father’s occupation and not his grandfather’s.

              suppose that such a model is appropriate and that the transition probability

              matrix is given by

                                               

 

               

 

               For such a population what fraction of people are middle class in the long run?

21. Define the Poisson process and find the expression for P_n(t).
22. Write short notes on the following:

(a). Stationary distribution

(b). Communicative sets and their equivalence property

(c). Periodic states

 

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      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]

 

 

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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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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.

1. 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

1. What proportion of people are lower class in the long run?
2. 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

 

1. a) Poisson Process
2. b) Irreducible Markov Chain

 

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