LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2003

ST-3800/S915 – STOCHASTIC PROCESSES

03.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL the questions.                                                                              (10×2=20 marks)

 

1. Define a stochastic process clearly explaining the time and state space.
2. Examine if a sequence of independent random variables possesses independent increment property.
3. Define a Markov chain. Give an example.
4. Let the transition probability matrix of a Markov chain with the state space S= {0,1,2,3} be P = . Find the periodicities of the states.
5. Define i) recurrence and ii) mean recurrence time of state i.
6. Describe a Poisson process.
7. Define current life and excess life associated with a renewal process.
8. Find the distribution of excess life if N(t) ~ P (l t).
9. If {X_n} is martingale with respect to {Y_n} , Show that E [X_n+k½Y₀ Y_1­­ ….Y_n] = X_n for all k .
10. Define Branching Process.

 

SECTION-B

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

 

11. State and establish Chapman – Kolmogorov equations satisfied by a discrete time Marhov – chain.
12. Show that d(i) = g cd {n ³ 1 | f> 0}.
13. Describe Yule process. Show that the marginal distirbution of the process is negative binomial with p = e^–bt if the initial size is greater than 1.
14. Describe a Birth – Death process and derive kolmogorov forward differential equation.
15. Show that the renewal function corresponding to the life time density

 

 

l² x e^–lx  ,   x > 0

f (x) =

0              ,    elsewhere

 

is given by      M (t) =

16. Define a renewal process. Show that renewal function M(t) satisfies the renewal equation.
17. Let Y₀ = 0, and Y₁, Y₂….be iid random variables with E (Y_k) = 0 and

E (Y) = s², k = 1, 2, ….

Let X₀ = 0 and X_n =  .

Show that {X_n}  is a martingale w.r.t {Y_n}.

18. Suppose the probability generating function of the off-spring distribution for a Branching process is f (s) = 0.1 + 0.4 s + 0.5 S². Obtain the extinction probability.

 

SECTION-C

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

 

19. i) Show that if i j, then d(i) = d(j). (8)
20. ii) Show that state i is recurrent if and only if . (12)
21. i) Prove that the three dimensional symmetric random walk on the set of integers is a transient Markov chain.          (15)
22. ii) Let {X_n, n ³ 0} be an irreducible FMC with doubly stochastic tpm. Show that the stationary probabilities are equal. (5)
23. i) Show that the stationary distribution for a single server queueing model is geometric, and also show that the distribution of waiting time is an exponential.                         (7+8)
24. ii) Derive Kolnogorov forward differential equations for Telephone trunking model. (5)
25. i) Let the renewal counting process be Poisson. Find the joint distribution of  and .  Deduce that the two random variables are independent.                                                        (10)
26. ii) State and prove elementary renewal theorem. (10)

 

 

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2006

                                                   ST 3806 – STOCHASTIC PROCESSES

 

 

Date & Time : 17-04-2006/AFTERNOON   Dept. No.                                                       Max. : 100 Marks

 

 

SECTION  A                           Answer all the questions                           10 ´ 2 = 20

 

1. Define stationary independent increment process.
2. Show that the square of a stochastic matrix is stochastic.
3. Suppose the one-step tpm is an identitiy matrix, show that the states are all recurrent.
4. Find a stationary disribution of an MC with one-step tpm

 

P =       0.3    0.7

0.7    0.3

5. For a Poisson process, find the covariance function.
6. Describe Pure birth process.
7. Define excess life and current life of a Renewal process.
8. For a martingale { X_n , n = 0.1.2,…}, show that E(X_n) = E(X_{n +1}), n = 0,1,2,…
9. Describe a branching process.
10. Define a covariance stationary process and give an example.

 

SECTION  B                               Answer any five questions                       5 ´ 8 = 40

 

11. For a stationary independent increment process, show that the variance of the marginal distribution is linear in the time parameter.
12. Define periodicity and show that it is a class property.
13. If the one step tpm of an irreducible finite state Markov Chain is symmetric, show that the stationary distribution is uniform.
14. Describe a Poisson Process and derive its marginal distribution.
15. For a linear growth process with immigration, find the average size of the population if the initial population is i units.
16. Derive the generating function relations satisfied by a Branching process.
17. If { X(t), t ³ 0} is a Brownian motion process, show that the distribution of (X(t₁),X(t₂)) is bivariate normal.
18. If the interoccurrence distribution of a Renewal process is exponential, find the distributions of i ) current life and ii ) excess life.

 

 

 

 

 

 

SECTION  C                              Answer any two questions                      2 ´ 20 = 40

 

19 a). State and establish Chapman-Kolmogorov equations satisfied by a Markov Chain.

b).   Illustrate Basic limit theorem with an example.

20 a). Describe Birth-Death process. Derive Kolmogorov backward equations satisfied by the Birth-Death process.

1. b) Describe telephone trunking model and find its stationary distribution.

 

21 .a) State and prove Elementary renewal theorem in Renewal theory.

1. b) Find the renewal function associated with a renewal process having the interoccurrence distribution with pdf

 

f(x) = l² x exp(-l x), x > 0, l > 0.

 

22 a) Let { X_n , n = 0.1.2,…}be a Branching process with X_{0 =} 1. Find the mean and variance of  X_n in terms of those of the offspring distribution.

1. b) Let { X_n , n = 0.1.2,…}be a covariance stationary process with zero mean and the covariance function R_x(v) . Find the best predictor of X_n+1 of the form aX_n, where a is a real constant.

 

 

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                     LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034    M.Sc. DEGREE EXAMINATION – STATISTICS

AB 26

THIRD SEMESTER – NOV 2006

         ST 3809 – STOCHASTIC PROCESSES

(Also equivalent to ST 3806)

 

 

Date & Time : 27-10-2006/9.00-12.00     Dept. No.                                                       Max. : 100 Marks

 

 

                   Section-A (10 × 2=20 marks)            

Answer ALL the questions

 

1. Define (a) Stationary increments

(b) Independent increments of a stochastic process

2. Define the period of a state of a Markov chain. Show that an absorbing state is recurrent.
3. Let j be a state for which f_jj⁽ⁿ⁾ = n/(2⁽ⁿ⁺¹⁾), n>0. Show that j is recurrent.
4. Write down the postulates for a birth and death process.
5. Define a Renewal process {N(t),t ≥ 0} and write down its renewal function.
6. Define a submartingale.

∞

7. Let {X_n, n≥0} be a Branching process with the off spring mean m<1. Evaluate E[ Σ X_n].

n=0

8. Define a Brownian motion process.
9. Show that a Markov Renewal process is a Markov Chain with one step transition probabilities.
10. Distinguish between wide-sense and strictly stationary processes.

 

Section-B

Answer any FIVE questions (5× 8 = 40 marks)

 

11. Show that a Markov chain is fully determined, when its initial distribution and the one step transition probabilities of the Markov chain are known.
12. Define a transient state and prove that transience is a class property. For any state i and a transient state j, prove that

∞

Σ p_ij⁽ⁿ⁾ <∞

n=1

13. Show that in a two dimensional symmetric random walk, all the states are recurrent.
14. Assume that a device fails when a cumulative effect of k shocks occur. If the shocks happen according to a Poisson process with the parameter λ, find the density function for the life T of the device.
15. Obtain the system of differential equations satisfied by the transition probabilities of the Yule process and calculate its transition  probabilities when the initial condition is

X(0) = N.

16. Derive the integral equation satisfied by the renewal function of a Renewal process.
17. Let {X(t) | t Є[0,∞)} be a standard Brownian motion process. Obtain the conditional distribution of X(t) given X(t₁)= α and X(t₂)=β, where t₁<t<t₂.
18. If {X_n} is a Branching process and φ _n (s) is the probability generating function of X_n, show that φ _n satisfies the relation φ _n (s)= φ _n-k (φ _k (s) ) for all k such that

k= 1,2,…,n.

 

 

 

Section-C

Answer any TWO questions (2×20 =40)

 

19.a.  Define a recurrent state. (2 marks)

1. State and prove the Chapman-Kolmogorov equations for a discrete time  discrete space Markov Chain.(10 marks)
2. Consider a random walk on the integers such that p_i,i+1 = p, p_i,i-1=q for all integers i (0<p<1,p+q=1). Determine p₀₀⁽ⁿ⁾.Also find the generating function of p₀₀⁽ⁿ⁾  .(8 marks)

20.a. Show that recurrence is a class property.(6 marks)

1. Show that states belonging to the same class have the same period.(6 marks)
2. If lm  p_jj⁽ⁿ⁾>0, show that j is positive recurrent and aperiodic.(8 marks)

n→∞

21.a   Stating the postulates for a birth and death process, derive Kolmogorov backward differential equations.(2+6 marks)

1. Obtain E[X(t)], where X(t) is a linear birth and death process.(12 marks)

22.a.   Define a discrete time Martingale and show that the means of the marginal distributions are equal. (8 marks)

1. State and prove the prediction theorem for minimum mean square error predictors.

(12 marks)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 44

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2007

ST 3809/3806/3800 – STOCHASTIC PROCESSES

 

 

 

Date & Time: 26/04/2007 / 9:00 – 12:00      Dept. No.                                       Max. : 100 Marks

 

 

 

SECTION-A (10 × 2 = 20 marks)

 

Answer ALL the questions. Each question carries TWO marks.

 

1. Define the term “Stochastic Process” with an example.

 

1. Let { X_n, n=0,1,2,…} be a Markov chain with state space S = {1,2,3} and transition probability matrix

 

 

1/2     1/4    1/4

P  =     2/3      0      1/3

3/5     2/5      0

 

Compute P[X₃=3 │X₁=1]

 

1. Explain the terns:

1. Recurrence time
2. Mean recurrence time.

 

1. For any state i and a transient state j, find the value of

lim p_ij⁽ⁿ⁾

n→∞

1. Under the condition X(0)=1 , obtain the mean of Yule process.
2. Define renewal function and find the same when the inter occurrence times are independent and identically distributed exponential.
3. Find the probability of ultimate extinction of a Branching Process with offspring distribution having the probability generating function 0.5s²+0.5.
4. Define a Brownian motion process.
5. Show that a Markov Renewal process is a Markov Chain with one step transition probabilities.
6. Give an example of a stationary process, which is not covariance stationary.

 

SECTION- B (5 × 8=40marks)

 

Answer any FIVE questions. Each question carries EIGHT marks

 

1. When do you say that two states of a Markov Chain communicate with each other? Show that communication is an equivalence relation.

 

1. Show that in a two dimensional symmetric random walk, all the states are recurrent.
2. State and establish Kolmogorov forward differential equations satisfied by a birth-death process.
3. Show that the sum of two independent Poisson processes is a Poisson process. Is the difference of two independent Poisson processes a Poisson process?
4. Derive the integral equation satisfied by the renewal function of a Renewal process.
5. Define:   (i) Sub martingale and (ii) Super martingale.  Give an example of a martingale which is not a Markov Chan.

 

1. Derive the recurrence relation satisfied by the probability generating function, where { X_n, n=0,1,2,… } is a Branching Process with X₀=1.
2. Show that an AR process can be represented by a MA process of infinite order.

 

SECTION – C (2 × 20=40)

Answer any TWO questions. Each question carries TWENTY marks

 

1. a)  State and prove Chapman- Kolmogorov equations for a discrete time Markov

chain.                                                                                             (8 marks)

 

1. Define a recurrent state j. Show that a state j is recurrent or transient according

as

∞

∑ p_jj⁽ⁿ⁾ = + ∞ or < ∞ ( in usual notation).                          (12 marks)

n=1

1. a)  State and prove the Basic limit theorem of Markov chains.          (12 marks)

1. If lim p_jj⁽ⁿ⁾ > 0, show that j is positive recurrent and aperiodic. (8 marks)

n→∞

1. a)  Obtain E[X(t)], where X(t) is a linear birth and death process.     (10 marks)

1. Define M│M│1 queue. Obtain E(W_Q) in this case, when the steady state solution exists. (10 marks)

1. a)  If {X_n, n=0,1,2,… } is the Galton-Watson Branching process, obtain E(X_n) and

Var(X_n).                                                                                         (12 marks)

1. State and prove the prediction theorem for minimum mean square error

predictors.                                                                                (8 marks)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 43

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2008

    ST 3809 / 3806 / 3800 – STOCHASTIC PROCESSES

 

 

 

Date : 29/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Section-A (10×2=20 marks)

Answer ALL the questions. Each question carries TWO marks.

 

1. Define a) Independent increments and b) Stationary increments of a stochastic process.
2. Let {X_n, n=0,1,2,…} be a Markov chain with state space S={1,2,3} and transition probability matrix

 

 

P =       ½     ¼      ¼

2/3     0         1/3

3/5     2/5      0

Compute P[X₁=2, X₂=3, X₃=1, X₄=3│X₀=3]

1. Define: a) Recurrent state b) Ergodic Markov chain.
2. If   lim  p_jj ⁽ⁿ⁾ >0, show that state j is positive recurrent.

n→∞

1. Derive the probability generating function of Yule process corresponding to the homogeneous case, when X(0)=1.
2. Write down the postulates for a birth and death process.
3. Give an example of naturally occurring process that can be modelled as a renewal process.
4. Define a semi Markov process.
5. Given the probability generating function to be f(s)= (as+b), a+b=1, a,b>0, determine the extinction probability if X(0)=1.
6. Give an example of stationary process, which is not covariance stationary.

 

Section-B (5×8=40 marks)

Answer any FIVE questions. Each question carries EIGHT marks.

 

1. Show that a Markov chain is fully determined, when its initial distribution and the one step transition probabilities of the Markov chain are known.
2. Show that in a three dimensional symmetric random work, all the states are transient
3. Examine for the existence of stationary distribution {П_j } for the Markov chain, whose transition probability matrix is specified by p₀₁=1,

p_ij = q_i    if j=i-1

= p_i    if j=i+1, where i=1,2,3… and p_i+q_i=1

1. Under the condition X(0) =1, obtain the mean and variance of Yule process.
2. For a M│M│1 queueing system, show that the queue length process is Markov. Obtain the distribution of the waiting time in steady state.Also find E(N) and E(W_Q).
3. Define the term Renewal process. Derive the renewal function, if the inter renewal times have density

f(x)=λ²e^{– λx}x, x>0, λ>0

 

 

1. Let Y₀=0 and Y₁,Y₂,Y₃,… be independent and identically distributed random

variables with E(Y_k)=0 and E(Y²_k)=σ² , k=1,2,3,… Let X(0)=0 and

n

Xn =  (  Σ Y_k )²  – n σ² . Show that {X_n } is a Martingale with respect to {Y_n}

k=1

1. Show that the process {X(t)} defined by X(t) = Acos wt + Bsin wt, where A and B are uncorrelated random variables each with mean zero and variance unity, with w a positive constant, is covariance stationary.

 

Section-C (2×20=40)

Answer any TWO equestions. Each question carries TWENTY marks.

 

19.a) State and prove the necessary and sufficient condition for the state i of a

Markov chain to be recurrent.                                 (10 marks)

1. b) For a one dimensional symmetric random walk on the set of integers, find

f₀₀⁽ⁿ⁾ .                                                                           (10 marks)

20a) Show that states belonging to the same class have the same period.(8 marks)

1. b) Define stationary distribution { П _j} of a Markov chain. Prove that, for an

irreducible Markov chain with a doubly stochastic matrix, { П _j} is given by

П _j = 1/M, j=1,2,…,M,where M is the number of states.                                                                                                                    (4+8 marks)

21.a) State Chapman-Kolmogorov equation for a continuous time Markov chain.

Deduce Kolmogorov forward and backward equations.         (10 marks).

1. b) Obtain E[X(t)], where X(t) is a linear birth and death process.(10 marks).

22.a) Derive the integral equation satisfied by the renewal function of a Renewal

process.                                                                                           (8 marks)

1. b) If {X_n, n=0,l,2,…} is the Galton-Watson branching process, find E(X_n) and

Var(X_n).                                                                                           (12 marks)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 25

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – November 2008

    ST 3809 / 3800 – STOCHASTIC PROCESSES

 

 

 

Date : 05-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART-A

Answer all questions:                                                                                      (10×2=20)

 

1. Define a Markov process.

 

2. Define recurrent state and transient state of a Markov chain.

 

3. Define a Martingale of the process {Xn} with respect to {Yn}.

 

4. Obtain E[X₁ + X₂ +…..+ X_N] where Xi, i=1, 2, 3,….. are i.i.d and independent of the

random variable N .

 

5. Let X₁, X₂ be independent exponentially distributed random variables parameters λ₁

and λ₂ respectively. Obtain P[min(X₁,X₂)>t] .

 

6. Messages arrive at the telegraph office in accordance with the laws of a Poisson

Process with mean rate of 3 messages per hour. What is the probability of getting no

message during morning hours from10 to 12?

 

7. Obtain the pgf of a Poisson process.

 

8. If X₁ and X₂ are independent random variables with distribution functions of F₁ and F₂ respectively. Write

an expression for the distribution function of X=X₁+X₂?

 

9. Obtain P[N(t)=k] in terms of the distribution functions of the life times for a renewal

Process?

 

10. Define a stationary process.

 

PART-B

Answer 5 questions:                                                                                        (5×8=40)

 

11) Consider the Markov chain with states 0,1,2 having the TPM

 

 

and  P[X₀ = i] = 1/3,  i = 1,2,3

Obtain i) P[X₂=0]

1. ii) P[X₂=0, X₁=2/ X₀=1]

iii) P[X₂=0, X₁=2, X_0­=1]                                                              (4+2+2)

 

12) Verify whether the Markov chain with TPM given below is ergodic

 

 

 

 

 

13) Show that for a renewal process in the usual notation,

M(t)= F(t) + F*M(t)

 

14) Prove that if {Xn} is a super martingale with respect to {Yn} then

1. i) E[Xn+k ç Y₀,Y₁,…..Yn ] ≤ Xn,
2. ii) E[Xn­] ≤ E[Xk], 0 ≤ k ≤ n

 

15) State the postulates of birth and death process. Obtain the forward differential

equations for a birth and death process.

 

16)  Obtain the Stationary distribution of a Markov chain with TPM

 

17) Consider the times {S_k} at which the changes of Poisson process X(t) occur. If

Si = T₀ + T₁ + … + T_i-1, i = 1,2,3,… obtain the joint distribution function of S₁,

S₂,……S_n given X(t) = n.

 

18) Show the periodicity is a class property.

 

PART-C

Answer 2 questions:                                                                                        (2 x 20=40)

 

19) a) Show that i is recurrent if and only if ∑P_ii n = ∞

1. b) Show that in a one dimensional symmetric random walk state 0 is recurrent.
2. c) if j is transient prove that for all i ∑P_ij n < ∞                                              (8+7+5)

 

20) a) State the postulates of a Poisson process and obtain the expression for P_n(t).

 

1. b) If X(t) has a Poisson process, u<t, k<n obtain P[X(u) = k çX(t) = n]             (12+8)

 

21) a) Obtain the renewal function corresponding to the lifetime density

f(x) = λ² x e ^{– λ x} ,  x ≥ 0

1. b) Let Y₀=0, Y₁, Y₂,….. be i.i.d with

E[Y_k] = 0    var[Y_k]=σ²     k=1,2,……

E[| Y_n |] < ∞   let X₀=0

Show that

1. i) X _n= Y_i
2. ii) X_n = (Y_i )² – nσ²

are martingales.                                                                                                 (10+5+5)

 

22) a) Derive the p.g.f of a branching process. Hence obtain the mean and variance of X_n.

 

1. b) Let the offspring distribution be P[ζ= i] = 1/3 , i = 0,1,2

Obtain the probability of extinction.

 

 

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    LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 42

THIRD SEMESTER – April 2009

ST 3809 – STOCHASTIC PROCESSES

 

 

 

Date & Time: 23/04/2009 / 1:00 – 4:00 Dept. No.                                                      Max. : 100 Marks

 

 

PART-A

             Answer all the questions:                                                                             (10 X 2 = 20)

1. 
   Define a stochastic process with an example.
2. Define a process with independent increments.

 

3. Show that communication between two states i and j satisfies transitive relation.

 

4. Define (i) transcient state (ii) recurrent state.

 

5. Define a Markov process.

 

6. Obtain the PGF of a Poisson process.

 

7. Define a renewal function. What is the relation between a renewal function and the

distribution functions of inter occurrence times?

 

8. When do you say that  is a martingale with respect to ?

 

9. What is a branching process?

 

10. What is the relationship between Poisson process and exponential distribution?

 

PART-B

            Answer any 5 questions:                                                                                    (5 X 8 = 40)

 

11. State and prove Chapman – Kolmogorov equation for a discrete time Markov chain.

 

12. Obtain the equation for in a Yule process with X(0) = 1.

 

13. Let and  be i.i.d random variables with mean 0 and variance.

Show that   is a martingale with respect to .

 

14. Show that the matrix of transition probabilities together with the initial distribution

completely specifies a Markov chain.

 

15. Show that the renewal function satisfies

 

 

16. Establish the relationship between Poisson process and Binomial distribution.

 

17. Obtain the stationary distribution for the Markov chain having transition probability

matrix

 

 

 

18. If a process has stationary independent increments and finite mean show that

 

where     and  .

 

PART-C

                 Answer any 2 questions:                                                                          (2 X 20 = 40)

 

19. a) State and prove the necessary and sufficient condition required by a state to be                               recurrent.

 

b.)  Verify whether state 0 is recurrent in a symmetric random walk in three dimensions.                                                                                                                                         (10+10)

 

20. a) State the postulates of a Poisson process. Obtain the expression for.

 

b.)  Obtain the distribution for waiting time of k arrivals for a Poisson process.                                                                                                                                                                    (15+5)

 

21. a) Obtain the generating function for a branching process. Hence obtain the mean and                        variance.

 

1.       b)  Let   be the probability that an individual in a generation generates k

off springs. If  obtain the probability of extinction.

(15+5)

22. a.) Obtain the renewal function corresponding to the lifetime density.

 

 

b.)  Show that the likelihood ratio forms a martingale.

 

c.)  Let be a martingale with respect to.  If  is a convex function with

 

show that   is a sub martingale with respect to .

                                                                                                                             (10+5+5)

                                        

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2012

ST 3812/3809 – STOCHASTIC PROCESSES

 

 

Date : 24-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

Section – A

       Answer all the questions:                                                                                         10 x 2 = 20 marks

1. Define convergence in quadratic mean.
2. Define periodicity and aperiodicity of a Markov chain.
3. Give an example for a reducible Markov chain.
4. Write the infinitesimal generator of a birth and death process.
5. Write any two applications of Poisson process.
6. Provide any two examples for renewal process.
7. Define a super martingale.
8. Define discrete time branching process.
9. Write a note on stationary process.
10. Write different types of stochastic processes.

                                                                                 

Section – B

Answer any five questions:                                                                                          5 x 8 = 40 marks

11. Explain (i) martingale   (ii) point process
12. Explain one-dimensional random walk.
13. (a) Show that a state i is recurrent if and only if _iiⁿ = .

• If i j and if i is recurrent show that j is recurrent.                                              (4+4)

14. Derive P_n(t) for the Yule process with X(0) = 1.
15. Derive the mean for a birth and death process if λ_n = nλ + a and μ_n = nμ with λ > 0 , μ >0 and a>0.
16. Explain (i) renewal function   (ii) excess life   (iii) current life  (iv) mean total life
17. Explain Markov branching process with three examples.
18. Write a note about (i) stationary process on the circle (ii) stationary Markov chains.

                                                                                  

 

Section – C

Answer any two questions:                                                                                        2 x 20 = 40 marks

 

19. (a) Show that state 0 is recurrent for a two dimensional random walk.

(b) Derive the basic limit theorem of Markov chains.                                                   (5+15)

 

20. For the gambler’s  ruin  on (n+1) states  with  P(X_n+1 = i+1  |  X_n = i ) = p ,

P(X_n+1 = i-1  |  X_n = i ) = q  and  0 and n are absorbing states , calculate u_i  = _i  (C₀ )  and  v _i ( C_n) .

21. (a) Derive the differential equations for pure birth process.

(b) Derive P_n(t)  for Yule process  with X(0) = N.                                                      (10 + 10)

22. (a) State and prove the basic renewal  theorem.

(b) Derive mean and variance of  branching process.                                                (10 + 10)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2012

ST 3812 – STOCHASTIC PROCESSES

 

 

Date : 03/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer all the questions:                                                                                          (10 x 2 = 20 Marks)

 

1. Define a point process.
2. Define n step transition probability.
3. Write any two basic properties of the period of a state.
4. If i « j and if i is recurrent then show that j is also recurrent.
5. Define mean recurrence time.
6. What is the infinitesimal generator of a birth and death process?
7. Define excess life and current life of a renewal process.
8. Define a sub martingale.
9. Write down the postulates of a birth and death process.
10. Write down any two examples for stationary process.

 

 

 

SECTION – B

Answer any Five questions:                                                                                     (5 x 8 = 40 Marks)

 

11. Explain (i) process with stationary independent increments (ii) Markov processes.
12. Explain spatially homogenous Markov chains.
13. Prove that a state i is recurrent if an only if

å  P_iiⁿ  =  ∞

14. For a two dimensional random walk, prove that å  P₀₀ⁿ  =  ∞
15. Determine stationary probability distribution for a random walk whose transition probability matrix is

 

0          1          0          0     . . .

q₁         0          p₁         0     . . .

0          q₂         0          p₂   . . .

P  =      .

.

.

 

 

 

 

 

-2-

 

16. Derive P_n (t) for a Poisson process.
17. Derive the expected value of a birth and death process with linear growth and immigration.
18. State and prove the basic renewal theorem.

 

 

 

SECTION – C

Answer any two questions:                                                                                      (2 x 20 = 40 Marks)

 

19. (a) State and prove the basic limit theorem of Markov chains.

(b)  Explain discrete renewal equation.                                                           (15 + 5)

 

20. (a) Derive the differential equations for a pure birth process.

(b)  Derive the Kolmogorov forward and backward differential equations of a birth and

death process.                                                                                          (10 +10)

 

21. (a) Explain renewal function, excess life, current life and mean total life.

(b)  If {X_t}is a renewal process with μ = E [X_t] < ∞ , then show that

lim 1/t M (t) = 1/μ  as t ® ∞                                                                   (8 + 12)

22. (a) Show that π is the smallest positive root of the equation j(s) = s

for a branching process.

(b)  Compute expectation and variance of branching process.                       (10 + 10)

 

 

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