LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
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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
- 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.
- Explain Lebesgue-Stieltjes measure with an example.
- Define counting measure with an example.
- State Borel- Cantelli Lemma.
- If h is B– measurable function, show that | h | is also B-measurable
function.
- What is induced probability space?
- If random variable X takes only positive integral values, show that
E(X) = P[ X ³ n]. - Define convergence in r-th mean.
- If Xn X and g is continuous, show that g(Xn) g (X).
- State Levy’s theorem.
Part B
Answer any five questions. 5 ´ 8 = 40
- If { Ai , i ³ 1) is a sequence of subsets of a set W, show that
Ai = (A i – A i – 1).
- Show that countable additivity of a set function with m(f) = 0 implies finite additivity of a set function.
- Prove that every finite measure is a s – finite measure. Is the converse true? Justify.
- Let f be B-measurable and if f = 0 a.e. [m], show that f dm = 0.
- State and establish additivity theorem of integral.
- State and establish Minkowski’s inequality.
- Show that Xn X implies Xn X. Is the converse true? Justify.
- If XnX, show that (Xn2 + Xn) (X2 + X).
Part C
Answer any two questions. 2 ´ 20 = 40
- a). State and establish extended monotone convergence theorem.
b). State and establish basic integration theorem. ( 12 + 8)
- 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)
- 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)
- 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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