Loyola College M.Sc. Statistics April 2006 Measure And Probability Theory Question Paper PDF Download

             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 An = ( -1/n, 1] if n is odd and

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

  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 Xn  X and g is continuous, show that g(Xn)  g (X).
  5. State Levy’s theorem.

Part B

Answer any five questions.                                                                     5 ´ 8 = 40

 

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

Ai = (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 Xn  X implies Xn  X. Is the converse true? Justify.
  3. If XnX, show that (Xn2 + Xn) (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 nk P[|X|>n] ® 0 as n ® µ.                                                         (10 + 10)

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

 

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