LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
FIRST SEMESTER – April 2009
YB 32
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)
11. Let x n be an increasing sequence of real numbers and let An   , xn  . What is the
connection between a.) lim sup x n and lim sup An b.) lim inf x n and lim inf An ?
n
n 
n 
n
12. 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
 hd exists then  hd   h d .


 

  is finite for 0  j  k , k  0 .
14. Show that if E X k is finite, then E X
j
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
0, x  0


x
F ( x)   1 3 1   3x 
is neither discrete nor continuous.
1

e

e
,
x

0
 2
2
17. State and prove Chebyshev’s inequality.
18. If Y1  Y2 a.e P , show that
a.) EY1 | g   EY2 | g  a.e P  and
b.) EY1 | X  x  EY2 | X  x a.e P .
1
SECTION C
Answer any TWO questions.
(2*20=40)
19. a.) Let An , n  1and Bn , n  1be two increasing sequences of sets defined
on , , P . If lim An  lim Bn then show that lim P An   lim PBn  .
n 
n 
n 
n 
b.) If  hd exists, show that  chd  c  hd where ‘c’ is a constant.



(6+14)
20. State and prove basic integration theorem.
21. a.) State and prove Weak law of large numbers.
b.) State and prove Minkowski’s inequality.
(10+10)
22. a.) 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
a .s .
X n 
 where Xn = max{Y1,Y2,…,Yn}.
(10+10).
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