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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc. DEGREE EXAMINATION - STATISTICS FIRST SEMESTER – November 2008 BA 20 ST 1809 - MEASURE AND PROBABILITY Date : 06-11-08 Time : 1:00 - 4:00 Dept. No. Max. : 100 Marks SECTION A Answer all questions. (10x2=20) 1. Give the definition of a -field. 1 n ,1 if n is odd 2. Let An . Find lim sup An and lim inf An . n n 1 1, if n is even n 3. Let 1 , 2 ,..., be the power set of and be a measure defined on . Define An n , n 1,2,... Verify whether is countably additive. 4. Define Signed measure. 5. Give an example for a -finite measure. 6. Give the relation between Lebesgue - Steiltje’s measure and the distribution function and hence show that Lebesgue measure is a particular case of Lebesgue - Steiltje’s measure. 7. Define Product measure. 8. State Radon – Nikodym theorem. 9. If Y1 Y2 a.e. P , then show that EY1 | X x EY2 | X x a.e. P . 10. State Weak law of large numbers. SECTION B Answer any FIVE questions. (5x8=40) 11. Let A1 , A2 ,..., An be subsets of Show that for each ‘n’ n A i i 1 A1 ( A2 A1c ) .... ( An Anc1 ... A1c ) . 12. Show that every field is a -field but the converse need not be true. 13. Explain the various ways of defining the integral of a borel measurable function. 14. State and prove Fatou’s lemma. 15. Show that . 16. State and prove Holder’s inequality. 17. Justify the following statement: “Existence of the higher order moments implies the existence of the lower order moments but the converse need not be true”. 18. State and prove monotone convergence theorem for conditional expectation given a -field. 1 SECTION C Answer any TWO questions. (2x20=40) 19. Let ‘h’ be a Borel measurable function such that hd exists. Define A hd , A . A Show that is countably additive on . In particular if h 0 , then show that is a measure on . 20. a.) Let ‘f ‘be a borel measurable function. If f 0 a.e. , then show that fd 0 . b.) State and prove monotone class theorem. (8+12) 21. a.) State and prove Chebyshev’s inequality. b.) A coin is tossed independently and indefinitely. Define the event A2n as in the 2nth toss equalization of head and tail occurs. Show that if the coin is biased, then the probability of A2n occurring infinitely often is zero and if the coin is unbiased, then the probability of A2n occurring infinitely often is one. (8+12) 22. a.) State and prove Kolmogorov’s strong law of large numbers. b.) Show that almost sure convergence need not imply convergence in quadratic mean. Further, show that quadratic mean convergence need not imply almost sure convergence. (10+10) ************** 2