LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 26

FIRST SEMESTER – APRIL 2006

                                       ST 1809 – MEASURE AND PROBABILITY THEORY

 

 

Date & Time : 25-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

Part A

Answer all the questions.                                                                            10 ´ 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 counting measure with an example.
3. State Borel- Cantelli Lemma.
4. If h is B– measurable function, show that | h | is also B-measurable

function.

1. What is induced probability space?
2. If random variable X takes only positive integral values, show that
   E(X) = P[ X ³ n].
3. Define convergence in r-th mean.
4. If X_n  X and g is continuous, show that g(X_n)  g (X).
5. State Levy’s theorem.

Part B

Answer any five questions.                                                                     5 ´ 8 = 40

 

1. If { A_i , i ³ 1) is a sequence of subsets of a set W, 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],  show that f dm = 0.
4. State and establish  additivity theorem of integral.

 

 

 

1. State and establish Minkowski’s inequality.
2. Show that X_n  X implies X_n  X. Is the converse true? Justify.
3. If X_nX, show that (X_n² + X_n) (X² + X).

 

Part C

Answer any two questions.                                                                   2 ´ 20 = 40

 

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

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

1. a). Let l (A) = dm;  A in the s – field Á, where fdm exists; thus l is a signed measure on Á. Show that l⁺(A) = f ⁺dm, l^– (A) = f ^– dm and |l|(A) = |f|dm.

b). State and establish Jordan – Hohn decomposition theorem.                    (8 + 12)

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

b). Let X be a random variable defined on the space (W, A, p) and E |X|^k < µ, k>0, Show that n^k P[|X|>n] ® 0 as n ® µ.                                                         (10 + 10)

1. a). Show that X_n  X implies X_n   X. Is the converse true? Justify.
   b). State and establish Lindberg Central limit theorem.                              (10 + 10)

 

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                      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AB 18

FIRST SEMESTER – NOV 2006

ST 1809 – MEASURE AND PROBABILITY THEORY

 

 

Date & Time : 28-10-2006/1.00-4.00    Dept. No.                                                       Max. : 100 Marks

 

 

 

Part A

Answer all the questions.                                                                            10 X 2 = 20

 

1.  Define minimal s – field.
2. Explain Lebesgue measure with an example.
3. What is a set function?
4. What is positive part and negative part of a borel measurable function?
5. State Randon – Nikodym theorem
6. Show that a random variable need not necessarily be a discrete or continuous type.
7. Define almost everywhere convergence.
8. State Holder’s inequality.
9. Describe a simple function with an example.
10. If X_n  X and g is continuous then show that g(X_n)  g (X).

 

Part B

 

Answer any five questions.                                                                           5 X 8 = 40

 

1. Show that countable additivity of a set function with m(f) = 0 implies finite additivity of a set function.
2. Show that a counting measure is a complete measure on a s – field.
3. Let F be the distribution function on R given by

0          if   x < -1

1 + x    if   -1 £ x < 0

F(x) =           2 + x²   if   0£ x < 2

9          if   x ³ 2.

If m is the Lebesgue – Stieltjes measure corresponding to F, compute the measure

of the set { x: ÷ x÷ + 2x² > 1}.

1. Let f be B-measurable and if f = 0 a.e. [m]. Then show that f dm = 0.
2. State and establish additivity theorem of integral.
3. State and establish Minkowski’s inequality.

 

 

1. If X_nX then show that (X_n² + X_n) (X² + X).
2. Describe Central Limit theorem and its purpose.

 

Part C

 

Answer any two questions.                                                                         2 X 20 = 40

 

1. a). 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}).

b). Show that a monotone class which a field is s – field.                          (10 +10)

1. a). State and establish basic integration theorem.

b). If hdm exists then show that ½hdm ½£ ïh ïdm                           (12 + 8)

1.  a). State and establish monotone class theorem.

b). If    X_n  X  then show that E½X_n½^r   E½X½^r  as n ® ¥. (12+ 8)

1. a). Show that Liapunov’s Central Limit theorem is a particular case of

Lindeberg’s Central Limit theorem.

b). State and establish Levy’s theorem.                                                       (8 + 12)

 

 

 

 

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