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.

 

 

Go To Main Page

Latest Govt Job & Exam Updates:

View Full List ...