LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

ZA 59

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – April 2009

MT 2961 – PROBABILITY THEORY AND STOCHASTIC PROCESSES

 

 

 

Date & Time: 02/05/2009 / 1:00 – 4:00  Dept. No.                                                Max. : 100 Marks

 

 

PART-A

Answer all questions                                                                                      (10 x 2 = 20)

 

1. Find the value of k such that, 0<x<1, zero, elsewhere represents the pdf of a random variable.
2. Find the mgf of a random variable X with probability mass function, x = 1,2,3,…
3. If A and B are two events and show that .
4. Define convergence in distribution of a sequence of random variables.
5. Define a stochastic process. What are the different ways of classifying a stochastic process?
6. Let X and Y be two independent random variables with N(10,4) and N(15,5) respectively. What is the distribution of 2X+3Y?
7. State Bonferronni’s inequality.
8. Define the period of a state. When do you it is aperiodic?
9. Patients arrive at a clinic according to Poisson process with mean rate of 2 per minute. What is the probability that no customer will arrive during a 2 minute interval?
10. The joint pdf of two random variables is, 0<<<1, zero, elsewhere.  Obtain the conditional pdf of  given .

 

PART – B

Answer any five questions                                                     ( 5 x 8 = 40)

 

11. Show that the distribution function satisfies, , and right continuity.
12. Derive the mgf of the normal distribution. Hence obtain mean and variance.
13. Show that binomial distribution tends to Poisson distribution under some conditions, to be stated.
14. _{Let X have a pdf f(x) = , 0 < x < ∞ and Y be another independent random variable with pdf g(y) = ,0 < y < ∞ .obtain the pdf of U= .}
15. _{Let {} be a Markov chain with states 1,2 and 3 and transition probability matrix}

                                                            

              _{If the initial distribution is (},,), find            

1. _i)
2.              ii)

iii)

 

 

16. _{Derive the expression for in a pure birth process with X(0) =0}
17. _{Let X and Y have the joint pdf,zero, elsewhere. Find E[Y|x].}
18. _{Prove that E(XY) = E(X) E(Y) when the random variables are continuous and independent. Is the converse true? Justify.}

_{PART – C}

_{Answer any two questions                                                 ( 2 x 20 = 40)}

 

19. a) State and prove Baye’s theorem.
20. b) Give an example to show that pair wise independence does not imply

independence of three events.

c). State and prove Boole’s Inequality.                                   ( 8 + 6 + 6)

20. a) State and prove Markov inequality. Deduce Chebyshev’s inequality.

b). Show that almost sure convergence implies convergence in        probability.  Is

the converse true?  Justify.

c). State and prove central limit theorem for a sequence of i.i.d        random variables.                                                ( 5 + 5 + 10 )

21. (a). State the postulates of Poisson process and derive an expression for .

(b). Obtain the pgf of a Poisson process. If  and  have

independent Poisson processes with parameters  and      respectively,

find.           ( 6 + 5+ 9)

22. a). Show that the Markov chain with the transition probability matrix

 

is ergodic.  Obtain the stationary distribution.

b). Show that communication is an equivalence relation.

c). Let  be the minimum obtained while throwing a die n-times.

Obtain the transition probability matrix.                         ( 12 + 4 + 4)

 

 

 

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