Download { } LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
FIRST SEMESTER – NOVEMBER 2007
ST 1809 - MEASURE AND PROBABILITY
Date : 27/10/2007
Time : 1:00 - 4:00
Dept. No.
BB 11
Max. : 100 Marks
SECTION A
Answer ALL questions
2 * 10 = 20
1. Let {An , n ≥ 1} be a sequence of numbers defined as follows.
2.
3.
4.
5.
6.
7.
  −1 
  n ,1 if n is odd

An =  
 − 1, 1  if n is even

n 
lim
lim
Find
sup An and
inf An
n→∞
n→∞
Show that every σ field is a field.
Define Counting measure.
Give the definition of a simple Borel measurable function.
State Radon – Nikodym theorem.
Give an example to show that two different random variables can have same distribution function.
If the random variable X takes only positive integer values then show that
∞
E ( X ) = ∑ P ( X ≥ n)
n =1
8. If Y1 ≤ Y2 then show that E ( Y1 | ġ ) ≤ E ( Y2 | ġ ).
9. Define Convergence in distribution of a sequence of random variables.
10. State the Central Limit Theorem of a sequence of random variables.
SECTION B
Answer any FIVE questions
5 * 8 = 40
11. Let {X n , n ≥ 1} be a sequence of real numbers and let An = ( -∞, Xn). What is the connection between
lim
lim
sup An and
sup X n
n→∞
n→∞
12. Let {An , n ≥ 1} and {Bn , n ≥ 1} be two increasing sequences of sets defined on
lim
lim
lim
lim
(Ω, ℑ, Ρ ) such that
An ⊂
Bn then show that
P ( An ) ≤
P ( Bn )
n→∞
n→∞
n→∞
n→∞
13. State and prove Fatou’s lemma.
14. State and prove Minkowski’s inequality.
15. a.) Define discrete and continuous random variable.
b.) Let X be a random variable with the distribution function
0 , xp0


1
, 0 ≤ x p1

4

1

Show that X is neither discrete nor continuous.
F ( x) = 
, 1≤ x p 2
2

1 + x − 2 , 2 ≤ x p 3
2
2

1 , x≥3
16. Let X be a random object defined on (Ω, ℑ, Ρ ) and let B ∈ ℑ . Show that
E ( IB | X = x) = P ( B | X =x ) a.e. [P].
17. Let Yn ≥ 0 v n ≥ 1 be a sequence of random variables such that Yn ↑ Y.
Show that E ( Yn | ġ ) ↑ E ( Y | ġ ) a.e. [P].
18. Let {Yn , n ≥ 1} be a sequence of iid random variables with the probability density function
a .s .
f ( y;θ ) = e −( y −θ ) , y ≥ θ . Let Xn = min ( X1,X2,…,Xn). Show that X n →
θ.
SECTION C
Answer any TWO questions
2 * 20 = 40
19. State and prove Basic Integration Theorem.
20. a.) State and prove Additivity theorem of integral of Borel measurable functions.
b.) Let µ be a measure and λ1 and λ2 be signed measures defined on the σ field ℑ
of subsets of Ω. Prove that if λ1 ⊥ µ and λ2 ⊥ µ , then (λ1 + λ2 ) ⊥ µ
(12+8)
21. a.) A coin is tossed independently and indefinitely. Define the event A2n as in the
2nth toss equalization of head and tail occurs. Show that P ( A2n occurring infinitely often) is zero
or one corresponding to
the coin being biased or unbiased respectively.
b.) State and prove Weak law of large numbers.
(14+6)
22. a.) Let (Ω, ℑ, Ρ ) be a probability space and let ġ be sub σ field of ℑ . Fix B ∈ ℑ .
Show that there is a function P(B | ġ ) : (Ω, ġ) → (R,B(R)) called the
probability of B | ġ such that P (C ∩ B ) = ∫ P ( B | g) dP ∀ c ∈ g . Further show that any two such
c
functions must coincide and P( B | ġ ) = E ( IB | ġ ) a.e. [P].
q .m .
a.s
b.) Show that X n →
X does not imply X n 
→ X . Also show that
q .m .
a.s
X n 
→ X does not imply X n →
X.
(12+8)
********
2