LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2004

ST 2800/S 815 – PROBABILITY THEORY

02.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

1. Show that if = 1, n = 1, 2, 3, …
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 (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).
6. If X is a random variable with P [examine whether E (X) exists.
7. State Glivenko – Cantelli theorem.
8. State Kolmogorov’s strong law of large number (SLLN).
9. If f(t) is the characteristic function of a random variable, examine if f(2t). f(t/2) is a haracteristic function.
10. Distinguish between the problem of law of large numbers and the central limit problem.

 

SECTION – B

 

Answer any FIVE questions                                                                        (5 ´ 8 = 40 marks)

 

11. The distribution function F of a random variable X is

           

0             if       x < -1

F (x) =          if      -1  £   x  < 0

if      0  £   x  < 1

1           if       1    £  x

Find Var (X)

 

12. If X is a non-negative random variable, show that E(X) < ¥ implies that
13. P [ X > n] ® 0 as n ® ¥.  Verify this result given that

f(x) = .

13. State and prove Minkowski’s inequality.
14. In the usual notation, prove that

.

15. Define convergence in quadratic mean and convergence in probability. Show that the former implies the latter.
16. Establish the following:
17. If X_n ® X with probability one, show that X_n ® X in probability.
18. Show that X_n ® X almost surely iff for every > 0,  is zero.
19. {X_n} is a sequence of independent random variables with common distribution function

 

 

0      if     x <  1

F(x) =

1-  if   1 £  x

Define Y_n = min (X₁, X₂ , … , X_n) .  Show that Y_n  converges almost surely to 1.

18. State and prove Kolmogorov zero – one law.

 

SECTION – C

 

Answer any TWO questions                                                                        (2 ´ 20 = 40 marks)

 

19. a) Let F be the range of X. If  and B F^C imply that P_X (B) = 0,  Show that P can

be uniquely defined on      (X), the s – field generated by X by the relation

P_X (B) = P {X Î B}.

1. b) Show that the random variable X is absolutely continuous, if its characteristic function f

is absolutely integrable over (- ¥,  ¥ ).  Find the density of X in terms of f.

20. a) State and prove Borel – Zero one law.
21. b) If {X_n, n ≥ 1} is a sequence of independent and identically distributed random

variables with common frequency function e^-x, x ≥ 0, prove that

.

21. a) State and prove Levy continuity theorem for a sequence of characteristic functions.
22. b) Use Levy continuity theorem to verify whether the independent sequence {X_n}

converges in distribution to a random variable, where X_n for each n, is uniformly

distributed over (-n, n).

22. a) Let {X_n} be a sequence of independent random variables with common frequency

function f(x) =, x ≥ 1.  Show that  does not coverage to zero with probability

one.

1. b) If X_n and Y_n are independent for each n, if X_n ® X, Y_n ® Y, both in distribution, prove

that ® (X² + Y²) in distribution.

1. c) Using central limit theorem for suitable exponential random variables, prove that

.

 

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