02.04.2004
1.00 - 4.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - STATISTICS
SECOND SEMESTER – APRIL 2004
ST 2800/S 815 - PROBABILITY THEORY
Max:100 marks
SECTION - A
(10  2 = 20 marks)
Answer ALL questions

1. Show that P (  A n )  1 if P (A n ) = 1, n = 1, 2, 3, ...
n 1
2. Define a random variable and its probability distribution.
3. Show that the probability distribution of a random variable is determined by its
distribution function.
4. Let F (x) = P ( X  x). Prove that F (.) is continuous to the right.
5. If X is a random variable with continuous distribution function F, obtain the probability
distribution of F (X).
1
6. If X is a random variable with P [ X  (1) k 2 k ]  k , k 1, 2, 3,..., examine whether E (X)
2
exists.
7. State Glivenko - Cantelli theorem.
8. State Kolmogorov's strong law of large number (SLLN).
9. If (t) is the characteristic function of a random variable, examine if (2t). (t/2) is a
haracteristic function.
10. Distinguish between the problem of law of large numbers and the central limit problem.
SECTION - B
(5  8 = 40 marks)
Answer any FIVE questions
11. The distribution function F of a random variable X is
F (x) =
0
if
x2
if
5
x2  3
if
5
1
if
x < -1
-1  x < 0
0  x <1
1
 x
Find Var (X)
12. If X is a non-negative random variable, show that E(X) <  implies that
n. P [ X > n]  0 as n  . Verify this result given that
2
f(x) = 3 , x 1 .
x
13. State and prove Minkowski's inequality.
1
14. In the usual notation, prove that

 X
P
 n E X 1 
n1

P X
 n .
n1
15. Define convergence in quadratic mean and convergence in probability. Show that the
former implies the latter.
16. Establish the following:
a) If Xn  X with probability one, show that Xn  X in probability.
b) Show that Xn  X almost surely iff for every  > 0, P lim sup X n  X  is


zero.
17. {Xn} is a sequence of independent random variables with common distribution function
0
if
x< 1
F(x) =
1
if 1  x
x
Define Yn = min (X1, X2 , ... , Xn) . Show that Yn converges almost surely to 1.
18. State and prove Kolmogorov zero - one law.
1-
SECTION - C
(2  20 = 40 marks)
Answer any TWO questions
19. a) Let F be the range of X. If   and B  FC imply that PX (B) = 0, Show that P can
be uniquely defined on
(X), the  - field generated by X by the relation
PX (B) = P {X  B}.
b) Show that the random variable X is absolutely continuous, if its characteristic function 
is absolutely integrable over (- ,  ). Find the density of X in terms of .
20. a) State and prove Borel - Zero one law.
b) If {Xn, n ≥ 1} is a sequence of independent and identically distributed random
variables with common frequency function e-x, x ≥ 0, prove that
Xn


P  lim
1  1 .
log n 

21. a) State and prove Levy continuity theorem for a sequence of characteristic functions.
b) Use Levy continuity theorem to verify whether the independent sequence {Xn}
converges in distribution to a random variable, where Xn for each n, is uniformly
distributed over (-n, n).
22. a) Let {Xn} be a sequence of independent random variables with common frequency
X
1
function f(x) = 2 , x ≥ 1. Show that n does not coverage to zero with probability
n
x
one.
b) If Xn and Yn are independent for each n, if Xn  X, Yn  Y, both in distribution, prove
that (X 2n  Yn2 )  (X2 + Y2) in distribution.
c) Using central limit theorem for suitable exponential random variables, prove that
n
1
e

n   (n  1)!
lim
0
t
t n1 dt 
1
.
2

2