LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 27

FIRST SEMESTER – APRIL 2007

ST 1809 – MEASURE AND PROBABILITY

 

 

 

Date & Time: 27/04/2007 / 1:00 – 4:00      Dept. No.                                       Max. : 100 Marks

 

 

 

Part A

Answer all the questions.                                                                            10 X 2 = 20

 

1. Define set of all real numbers as follows. Let A_n = ( -1/n, 1] if n is odd and

A_n = ( -1, 1/n] if n is even. Find lim sup A_n and lim inf A_n?

1. Explain Lebesgue-Stieltjes measure with an example.
2. Define counter measure with an example.
3. State Borel- Cantelli Lemma.
4. If h is B– measurable function then show that | h | is also B-measurable

function.

1. What is induced probability space?
2. If random variable X takes only positive integral values then show that

E(X) = P[ X ³ n].

1. Define convergence in r-th mean.
2. Explain Fatou’s lemma.
3. State Jordan-Hahn decomposition theorem.

Part B

Answer any five questions.                                                                     5 X 8 = 40

 

1. If { A_i , i ³ 1) is a sequence of subsets of a set W then show that

A_i = (A _i  – A _{i – 1}).

1. Show that countable additivity of a set function with m(f) = 0 implies finite additivity of a set function.
2. Prove that every finite measure is a s – finite measure. Is the converse true? Justify.
3. Let f be B-measurable and if f = 0 a.e. [m] then show that f dm = 0.
4. State and establish additivity theorem of integral.

 

 

1. State and establish Minkowski’s inequality.
2. If X_nX then show that (X_n² + X_n) (X² + X).
3. State and establish Levy’s theorem.

Part C

Answer any two questions.                                                                   2 X 20 = 40

 

1. a). State and establish extended monotone convergence theorem.

b). State and establish basic integration theorem.                                                                  ( 12 + 8)

1. a). Let Á₀ be a field of subsets of W. Let P be a probability measure on

Á_0. Let     { A_n , n ³ 1}and  {B_n,  n ³ 1} be two increasing sequence of sets such that

lim (A_n) Ì lim (B_n). Then prove that lim P(A_n) £ lim P(B_n)

b). Define absolute continuity of measures. Show that l << m if and only if  ½l½ << m.

(8 + 12)

1.  a). Show that X_n  X implies X_n   X. Is the converse true? Justify.

If X_n  C then show that X_n  C, where C is constant.

 

b). State and establish Lindberg Central limit theorem.                                                         (8 + 12)

1. a). If hdm exists and C є R then show that Chdm = Cdm.

b). If X_n  X and g is continuous then show that g(X_n)  g (X).

(12 + 8)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 20

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – November 2008

    ST 1809 – MEASURE AND PROBABILITY

 

 

 

Date : 06-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION A

Answer all questions.                                                                                    (10×2=20)

 

1. Give the definition of a -field.
2. Let. Find and.
3. Let, be the power set of  and be a measure defined on. Define 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  a.e., then show that  a.e..
10. State Weak law of large numbers.

 

 

SECTION B

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

 

1. Let be subsets of W. Show that for each ‘n’  .
2. Show that every field is a -field but the converse need not be true.
3. Explain the various ways of defining the integral of a borel measurable function.
4. State and prove Fatou’s lemma.
5. Show that.
6. State and prove Holder’s inequality.
7. 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”.

1. State and prove monotone convergence theorem for conditional expectation given a -field.

 

 

 

SECTION C

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

 

1. Let ‘h’ be a Borel measurable function such that exists. Define , . Show that is countably additive on. In particular if, then show that is a measure on.
2. a.) Let ‘f ‘be a borel measurable function. If a.e., then show that.

b.) State and prove monotone class theorem.                                     (8+12)

1. a.) State and prove Chebyshev’s inequality.

b.) A coin is tossed independently and indefinitely. Define the event A_2n as in the 2n^th toss equalization of head and tail occurs. Show that if the coin is biased, then the probability of A_2n occurring infinitely often is zero and if the coin is unbiased, then the probability of A_2n occurring infinitely often is one.

(8+12)

1. 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)

 

 

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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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