Download ST 1809 - Loyola College

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
NO 32
FIRST SEMESTER – APRIL 2008
ST 1809 - MEASURE AND PROBABILITY THEORY
Date : 30-04-08
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
SECTION A
Answer ALL questions
2 *10 = 20
C
lim
 lim

inf Acn
1. Show that 
sup An  =
n
n  

2. Define : Sigma field
3. Mention any two properties of set functions.
4. If
 hd
exists, then show that
 hd    h d 



5. Define Singular measure.
6. State the theorem of total probability.
7. If E (Xk) is finite, k > 0, then show that E (Xj) is finite for 0 < j < k.
3
, x  1. Find V(X1 + X2).
8. Let X1 and X2 be two iid random variables with pdf f ( x) 
4
2x
9. Show that E ( E ( Y | ġ ) ) = E (Y).
10. Define Convergence in rth mean of a sequence of random variables.
SECTION B
Answer any FIVE questions
5 * 8 = 40.
11. Show that finite additivity of a set function need not imply countable additivity.
12. Consider the following distribution function.
 0 if x  1
1  x if 1  x  0

F ( x)  
2
2  x if 0  x  2

9 if x  2
If μ is a Lebesgue measure corresponding to F, compute the measure of
 1
a.) 0,   1,2
b.) x : x  2 x2  1
 2


13. State and prove the Order Preservation Property of integral of Borel measurable functions.
14. 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.
15. State and prove Borel – Cantelli lemma.
16. Derive the defining equation of conditional expectation of a random variable given a σ field.
1
17. Let Y1,Y2,…,Yn be n iid random variables from U(0,θ). Define
a.s

Xn = max (Y1,Y2,…,Yn). Show that X n 
18. State and prove the Weak law of large numbers.
SECTION C
Answer any TWO questions.
2 * 20 = 40
19. 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  hd exists,

then show that  chd = c  hd , v c € R.

(8+12)

20. a.) State and prove the extended monotone convergence theorem for a sequence
of Borel measurable functions.
b.) If X = (X1,X2,…,Xn) has a density f(.) and each Xi has a density fi, i= 1,2,…,n ,
then show that X1,X2,…,Xn are independent if and only if f ( x1 , x2 ,..., xn ) 
n
 f j (x j )
j 1
a.e. [λ] except possibly on the Borel set of Rn with Lebesgue measure zero.
(12+8)
21. 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].
. m.
p.

X implies X n 
X but the converse is not true.
b.) Show that X n q
(10+10)
22. a.) State and prove Levy inversion theorem.
b.) Using Central limit theorem for suitable Poisson variable, prove that
lim
n
e n
k
 n

k 0 k !
 12 .
(12+8)
2
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