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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