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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 32

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2008

ST 1809 – MEASURE AND PROBABILITY THEORY

 

 

 

Date : 30-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION A

 

Answer ALL questions                                                                     2 *10 = 20

 

1. Show that  =
2. Define : Sigma field
3. Mention any two properties of set functions.
4. If   exists, then show that
5. Define Singular measure.
6. State the theorem of total probability.
7. If E (X^k) is finite, k > 0, then show that E (X^j) is finite for 0 < j < k.
8. Let X₁ and X₂­ be two iid random variables with pdf  . Find V(X₁ + X₂).
9. Show that E ( E ( Y | ġ ) ) = E (Y).
10. Define Convergence in r^th  mean of a sequence of random variables.

 

SECTION B

Answer any FIVE questions                                                                                    5 * 8 = 40.

 

1. Show that finite additivity of a set function need not imply countable additivity.
2. Consider the following distribution function.

If μ is a Lebesgue measure corresponding to F, compute the measure of

a.)          b.)

 

1. State and prove the Order Preservation Property of integral of Borel measurable functions.
2. Let μ be a measure and λ be a singed measure defined on the σ field  of subsets of Ω. Show that λ << μ   if and only if | λ | << μ.
3. State and prove Borel – Cantelli lemma.
4. Derive the defining equation of conditional expectation of a random variable given a σ field.

 

 

1. Let Y₁,Y₂,…,Y_n be n iid random variables from U(0,θ). Define

X_n = max (Y₁,Y₂,…,Y_n). Show that

 

1. State and prove the Weak law of large numbers.

 

SECTION C

Answer any TWO questions.                                                                         2 * 20 = 40

 

1.  a.) Show that every finite measure is a σ finite measure but the converse need not

be true.

b.) Let h be a Borel measurable function defined on. If   exists,

then show that  = , v  c € R.                                              (8+12)

1. a.) State and prove the extended monotone convergence theorem for a sequence

of Borel measurable functions.

b.) If X = (X₁,X₂,…,X_n) has a density f(.) and each X_i has a density f_i, i= 1,2,…,n ,

then show that X₁,X₂,…,X_n are independent if and only if

a.e. [λ] except possibly on the Borel set of Rⁿ with Lebesgue measure zero.

(12+8)

 

1. a.) If Z is ġ measurable and Y and YZ are integrable, then show that

E ( YZ | ġ ) = Z E (Y | ġ) a.e. [P].

b.) Show that  implies  but the converse is not true.

(10+10)

1.  a.) State and prove Levy inversion theorem.

b.) Using Central limit theorem for suitable Poisson variable, prove that

.                                                                           (12+8)

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 32

FIRST SEMESTER – April 2009

ST 1809 – MEASURE AND PROBABILITY THEORY

 

 

 

Date & Time: 25/04/2009 / 1:00 – 4:00  Dept. No.                                                     Max. : 100 Marks

 

 

SECTION A

      Answer all questions.                                                                              (10  x 2 = 20)

 

1. Define limit inferior of a sequence of sets.
2. Mention the difference between a field and a σ – field.
3. Give an example for counting measure.
4. Define Minimal σ – field.
5. Show that a Borel set need not be an interval.
6. Define Signed measure.
7. State Radon – Nikodym theorem.
8. Show that the Lebesgue measure of any interval is its length.
9. State Borel-Cantelli lemma.
10. Mention the various types of convergence.

 

SECTION B

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

 

1. Let be an increasing sequence of real numbers and let. What is the connection between a.) and b.) and ?

 

1. Show that every finite measure is a σ – finite measure but the converse need not be true.

 

13. State and prove the order preservation property of integrals and hence show that if exists then.

 

14. Show that ifis finite, then  is finite for.

 

15. State and prove Monotone convergence theorem for conditional expectation given a random object.

 

16. Show that the random variable X having the distribution function is neither discrete nor continuous.
17. State and prove Chebyshev’s inequality.

 

18. If e , show that

a.)  a.e   and

b.)  a.e .

 

                                                     

SECTION C

 

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

 

19. ) Let andbe two increasing sequences of sets defined
    on. If  then show that.

b.) If  exists, show that where ‘c’ is a constant.
(6+14)

 

20. State and prove basic integration theorem.

 

21. ) State and prove Weak law of large numbers.

 

b.) State and prove Minkowski’s inequality.                                      (10+10)

 

22. ) Derive the defining equations of the conditional expectation given a random
    object and given a -field.

 

b.) Let Y₁,Y₂,…,Yn be iid random variables from U(0,θ), θ > 0. Show that
where X_n = max{Y₁,Y₂,…,Y_n}.                              (10+10).

 

 

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