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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