LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 32

FIRST SEMESTER – April 2009

ST 1809 – MEASURE AND PROBABILITY THEORY

 

 

 

Date & Time: 25/04/2009 / 1:00 – 4:00  Dept. No.                                                     Max. : 100 Marks

 

 

SECTION A

      Answer all questions.                                                                              (10  x 2 = 20)

 

1. Define limit inferior of a sequence of sets.
2. Mention the difference between a field and a σ – field.
3. Give an example for counting measure.
4. Define Minimal σ – field.
5. Show that a Borel set need not be an interval.
6. Define Signed measure.
7. State Radon – Nikodym theorem.
8. Show that the Lebesgue measure of any interval is its length.
9. State Borel-Cantelli lemma.
10. Mention the various types of convergence.

 

SECTION B

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

 

1. Let be an increasing sequence of real numbers and let. What is the connection between a.) and b.) and ?

 

1. Show that every finite measure is a σ – finite measure but the converse need not be true.

 

13. State and prove the order preservation property of integrals and hence show that if exists then.

 

14. Show that ifis finite, then  is finite for.

 

15. State and prove Monotone convergence theorem for conditional expectation given a random object.

 

16. Show that the random variable X having the distribution function is neither discrete nor continuous.
17. State and prove Chebyshev’s inequality.

 

18. If e , show that

a.)  a.e   and

b.)  a.e .

 

                                                     

SECTION C

 

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

 

19. ) Let andbe two increasing sequences of sets defined
    on. If  then show that.

b.) If  exists, show that where ‘c’ is a constant.
(6+14)

 

20. State and prove basic integration theorem.

 

21. ) State and prove Weak law of large numbers.

 

b.) State and prove Minkowski’s inequality.                                      (10+10)

 

22. ) Derive the defining equations of the conditional expectation given a random
    object and given a -field.

 

b.) Let Y₁,Y₂,…,Yn be iid random variables from U(0,θ), θ > 0. Show that
where X_n = max{Y₁,Y₂,…,Y_n}.                              (10+10).

 

 

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