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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