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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