Loyola College B.Sc. Statistics Nov 2003 Applied Stochastic Process Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

FIFTH SEMESTER – NOVEMBER 2003

ST-5400/STA400 – APPLIED STOCHASTIC PROCESS

12.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION-A

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

 

  1. Define a stochastic process with an example.
  2. Define Bernoulli process with an example.
  3. Give an example of a continuous time, discrete space stochastic process.
  4. When do you say a process has stationary independent increments?
  5. Define a Markov Chain.
  6. Explain a doubly stochastic matrix.
  7. Define the periodicity of a state i of a Markov Chain.
  8. When the state i is said to be recurrent?
  9. Explain the one dimensional random walk with an example.
  10. Define a Martingale.

 

SECTION-B

 

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

 

  1. Let {Xn, n=0, 1, 2, …} be a sequence of iid r.v’s with common distribution P(Xo = i) = pI, iÎS, pI>0, . Prove that {Xn, n = 0. 1,2, ….} is a Markov chain.
  2. Prove that the state i of a Markov chain is recurrent iff .
  3. Show that the state ‘0’ of a one-dimensional symmetric random walk is recurrent.
  4. Explain a counting process with an example.
  5. Explain a Poisson process with an example.
  6. If {X1 (t), t Î (o, ¥,} and {X2 (t), t Î (o, ¥,} one two independent Poisson processes with parameters l1 and l2, Show that the distribution of X1 (t) / (X1 (t) + X2(t) = n follows B

.

  1. Explain in detail the generalization of a Poisson process.
  2. Let {Xt, t Î T} be a process with stationary independent increments when T – 0,1,2….. show that the process is a Markov process.

 

 

 

 

 

SECTION-C

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

 

  1. a) Show that a communication is an equivalence relation.
  2. b) Let{Xn, n ≥ 0} be a Markov Chain with state space S = {0,1,2,3,4} and transition

probability matrix      p=.     Find the equivalence classes.

  1. Show that for a Poisson process with distribution of x(t) is given by P{x(t) = m} =

 

  1. Explain the two dimensional symmetric random walk. Also prove that the state ‘0’ is recurrent.
  2. a) Let Y = 0 andY1, Y2 .. be iid r.r’s with E(Yk) = 0 and E(Yk2) = s2 ” k = 1,2, ….

Let Xn = .  Show that {Xn, n≥ 0 } is a martingale w.r.t {Yn, n ≥ 0}.

  1. b) Write short notes on Renewal process.

 

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