LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2007

ST 2805 / 2800 – PROBABILITY THEORY

 

 

 

Date & Time: 27/04/2007 / 1:00 – 4:00 Dept. No.                                            Max. : 100 Marks

 

 

Section – A

Answer all the Questions                                                         10 x 2 = 20

 

1. With reference to tossing a regular coin once and noting the outcome, identify completely all the elements of the probability space. (W,       , P).
2. Show that the limit of any convergent sequence of events is an event.
3. Define a random variable and its probability distribution.
4. If X is a random variable with continuous distribution functions F, obtain the probability of distribution of F(X).
5. Write down any two properties of the distribution function of a random vector (X,Y).
6. If X² and Y² are independent, are X and Y independent?
7. Define (i) convergence in quadratic mean and   (ii) convergence in distribution for a sequence of random variables.
8. If  f  is the characteristic function (CF) of a random variable X, find the CF of (3X+2).
9. State Kolmogorov’s strong law of large numbers(SLLN).
10. State Linde berg – Feller central limit theorem.

 

SECTION – B

Answer any FIVE questions.                                                  5 x 8 = 40

 

1. If X  and Y are independent, show that the characteristic function of X+Y is the product of their characteristic functions.  Is the converse true?  Justify.
2. State and prove Minkowski’s inequality.
3. Show that convergence in probability implies convergence in distribution.
4. State and prove Borel zero – one law.
5. Find the variance of Y, if the conditional characteristic function of Y given X=x is      and X has frequency function

for x  ³ 1

f (x) =

0,  otherwise

 

1. Show that X_n  ® X in probability if and only if every subsequence of {Xn} contains a further subsequence, with convergence almost surely.
2. Using the central limit theorem for suitable Poison random variables, prove that

=

 

 

1. Deduce Liapounov theorem from Lindeberg – Feller theorem.

 

 

Section – C

Answer any TWO questions                                                   2 x 20 = 40

 

19. a) Show that the probability distribution of a random variable is determined by its

distribution function.  Is the converse true?                                        (8  marks)

1. b) Show that the vector X = (X_1, X₂, …, X_p)  is a random vector if and only X_i,  i=1,2,…p is a real

random variable.                                                                                                          (8 marks)

1. c) The distribution function of a random variable X is given by

 

0       if  x < 0

F(x) =          if 0 £ x < 1

1       if  1 £ x < ¥

 

Obtain E(X).                                                                                       (4 marks)

 

20. a) State and prove Kolmogorov zero – one law for a sequence of independent
    random variables. (10 marks)
21. b) If {X_n , n ³ 1} is a sequence of independent and identically distributed random
    variables with common frequency function e^-x,  x  > 0,

prove that P[lim sup   ]=1

(10 marks)

21. a) State and prove Kolmogorov three series theorem for almost sure convergence
    of the series S X_n of independent random variables.                         (12 marks)
22. b) Show that convergence in quadratic mean implies convergence in probability.
    Illustrate by an example that the converse is not true.                         (8 marks)

 

22. a) State and prove Levy continuity theorem for a sequence of characteristic
    functions.           (10 marks)

 

1. b) State and prove Inversion theorem.           (10 marks)

 

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