10.11.2003
1.00 - 4.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
M.Sc., DEGREE EXAMINATION - MATHEMATICS
THIRD SEMESTER – NOVEMBER 2003
ST 3951 / S 972 - MATHEMATICAL STATISTICS - I
Max:100 marks
SECTION-A
Answer ALL questions.
(10x2=20 marks)
1. Illustrate that pairwise independence does not imply mutual independence of random
events.
2. Prove that every distribution function is continuous atleast from the left.
3. Show that if the probability of a random event equals zero, it does not follow that this
event is impossible. Similarly prove that if the probability of a random event equals one,
its does not follow that this event is sure.
4. Define truncated distribution of a random variable X and given an example.
5. Give two examples of random variables for which expected value does not exist.
6. Define convergence in law of a sequence of random variables and give an example.
7. Show that if the moment of order k of a random variable X exists, then
lim a k P  X  a   0, where a > 0.
a 
8. State the theorem of Bochner, giving necessary and sufficient conditions for a function
 (t ) to be a characteristic function.
9. The characteristic function of the random variable X is given by  (t ) = exp (t 2 / 2).
Find the density function of this random variable.
10. State Lindeberg - Levy Central Limit Theorem.
SECTION-B
Answer any FIVE questions.
(5x8=40 marks)
11. Let {An}, n =1, 2, ….., be a non increasing sequence of events and let A be their product.
Then show that P (A) = lim P ( An ) .
n
12. Show that the conditional probability satisfies the axioms of the theory of probability.
13. a) State and prove Bayes theorem.
b) Illustrate the application of Bayes formula.
14. State and prove a necessary and sufficient condition for the independence of the random
variables X and Y of the discrete type.
15. If a random variable has a symmetric distribution and its expected value exists, then show
that this expected value equals the center of symmetry. Hence show that for a symmetric
distribution the central moments of odd orders (if they exist) are equal to zero.
16. a) If not all the moments exist, then show that those moments that do exists fail to
determine the distribution function F (x).
b) Define convergence in rth mean and given an example.
17. The random variables X and Y have the joint density given by
 x 2  2 xy  2 y 2 
1
 . Compute the coefficient of correlation.
exp  
2
2


18. Define the t, chi-square and F distributions.
f (x,y) =
1
SECTION-C
Answer any TWO questions.
(2x20=40 marks)
19. a) State and prove Lapunov inequality concerning absolute moments.
(10)
b) Show that the expected value of the product of an arbitrary finite number of
independent random variables, whose expected values exist, equals the product of the
expected values of these variables.
(4)
c) Show that the covariance of two independent random variables equals zero. Is the
converse true? Justify your answer.
(6)
20. a) State and prove Levy Inversion Theorem concerning the determination of the
distribution function by the characteristic function.
(14)
b) Prove that the probability function of the Poisson distribution can be obtained as the
limit of a sequence of probability functions of the binomial distribution.
(6)
21. a) Show that for n  2, the binomial distribution can be obtained from the zero-one
distribution.
(4)
b) Examine the additive property for Gamma random variables.
(6)
c) State and prove Bernoulli's weak law of large numbers.
(10)
22. a) State and prove the Chebyshev inequality. Show that in the class of random variables
whose second order moment exists, one cannot obtain a better inequality than the
Chebyshev inequality.
(10)
b) If the  th moment m of a random variable exists, then show that m can be expressed
in terms of the  th derivative of the characteristic function  (t ) of this random variable
at t = 0.
c) Define the strong law of large numbers.
(8)
(2)

2