LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2012

ST 1814/1809 – MEASURE AND PROBABILITY

 

 

Date : 25-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

Answer all the questions                                                                                                       (10×2=20)

1. Define Increasing and Decreasing sequence of sets
2. Define Field
3. Define Monotone class
4. Define Borel σ-field
5. Define Measure
6. Define Random variable
7. State Chebyshev’s Inequality
8. State Minkowski’s Inequality
9. Establish: E[log(X)]≤log[E(X)]
10. Define Convergence in Distribution

SECTION B

Answer any five questions                                                                                                    (5×8=40)

11. i) Establish: If A₁,A₂,A₃,. . . , A_n be subsets of Ω, then
12. ii) If {A_n, n ≥1} is an increasing sequence of subsets of Ω then
13. State and Establish Cauchy-Schwartz Inequality
14. Establish: Every σ-field is a field but the converse is not true
15. Establish: If and   then
16. Prove by an Example: X² and Y² are independent need not imply X and Y are independent
17. i) Establish: If E[h(X)] exist then E[h(X)] = E[E{h(X)|y}]                                      (6)
18. ii) Define Jenson’s Inequality (2)

 

 

 

17. Find the density function of a distribution whose characteristic function is given below
18. State Lindeberg-Feller Central limit theorem and hence prove Liapounov’s Central Limit therorem

SECTION C

Answer any two questions                                                                                              (2×20=40)

19. State and prove Basic Integration theorem
20. i) State and Prove Monotone Convergence Theorem                                                             (10)
21. ii) The Minimal σ-field generated by the class of all open intervals is a Borel σ-field         (10)
22. i)  State and Establish Minkowski’s Inequality                                                                     (10)
23. ii) Let µ be a finitely additive set function on a field F of subsets of Ω. Further let µ is

continuous from above at Φ F    , then µ is countably additive on F                                       (10)

22. i) State and prove Inversion theorem  on characteristic function                                         (10)
23. ii) State and Establish Lindeberg-Levy Central limit theorem                                                              (10)

 

 

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