LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2006

                       ST 2902 – PROBABILITY THEORY AND STOCHASTIC PROCESSES

 

 

Date & Time : 28-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

Answer ALL the questions                                                                                                              (10 ´ 2 = 20)

1. Define probability by classical method.
2. Give an example for a discrete probability distribution.
3. Define an induced probability space.
4. State the properties of a distribution function.
5. Define the distributed function of a continuous random variable.
6. Write the formula to find the conditional mean and variance of Y given X = x.
7. What do you mean by a Markov matrix? Give an example
8. Write a note on one-dimensional random walk.
9. Define (i) recurrence of a state           (ii) periodicity of a state

10. Define renewal function.

PART – B

Answer any FIVE questions.                                                                                                             (5 ´ 8 = 40)

11. State and prove Boole’s inequality.
12. Explain multinomial distribution with an example.
13. Given the dF

F(x) =       0     ,  x < – 1

 

=    ,  -1

=      1        ,  1

 

compute (a) P(-1/2 < X  1/2)         (b) P(X = 0)    (c) P(X = 1)       (d) P (2 < X  3).

 

14. Let X have the pdf f(x) = 2x,  0 < x < 1, zero elsewhere. Find the dF and p.d.f. of Y = X².

 

15. (a) When is a Markov process called a Markov chain?

(b) Show that communication is an equivalence relation.                                                              (2 + 6)

 

16. A Markov chain on states {0,1,2,3,4,5} has t.p.m.

 

Find the equivalence classes.

 

17. Find the periodicity of the various states for a Markov chain with t.p.m.

 

 

 

 

18. Derive the differential equations for a pure birth process clearly stating the postulates.

 

PART – C

Answer any TWO questions.                                                                                                           (2 ´ 20 = 40)

 

19. (a) The probabilities that the independent events A,B and C will occur are ¼, ½ , ¼ respectively.
    What is the probability that at least one of the three events will occur?

 

• Find the mean and variance of the distribution that has the dF

 

F(x)  =  0         ,  x < 0

=  x/8      ,  0  £  x < 2

=  x²/16   ,  2  £  x < 4

=  1         ,  4  £  x                                                                                                     (5 + 15)

 

20. If X₁ and X₂ have the joint p.d.f.

 

f(x₁,x₂₎ =

 

find     (i) marginal pdf of X₁ and X_2.

                                                (ii) conditional pdf  of X₂ given X₁ = x₁ and X₁ given X₂ = x₂.

(iii) find the conditional mean and variance of X₂ given X₁ = x₁ and

X₁ given X₂ = x_2.                                                                                                                  (4 + 4 + 12)                      

 

21. Derive a Poisson process clearly stating the postulates.

 

22. Derive the backward and forward Kolmogorov differential equations for a

birth and death process clearly stating the postulates.

 

 

 

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