LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2003

ST 3951 / S 972 – MATHEMATICAL STATISTICS – I

10.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL questions.                                                                                   (10×2=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

where a > 0.

8. State the theorem of Bochner, giving necessary and sufficient conditions for a function to be a characteristic function.
9. The characteristic function of the random variable X is given by = exp Find the density function of this random variable.
10. State Lindeberg – Levy Central  Limit

SECTION-B

Answer any FIVE questions.                                                                           (5×8=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) = .
12. Show that the conditional probability satisfies the axioms of the theory of probability.
13. a) State and prove Bayes
14. b) Illustrate the application of Bayes
15. State and prove a necessary and sufficient condition for the independence of the random variables X and Y of the discrete type.
16. 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.
17. a) If not all the moments exist, then show that those moments that do exists fail to determine the distribution function F (x).
18. b) Define convergence in r^th mean and given an example.
19. The random variables X and Y have the joint density given by

f (x,y) = .   Compute the coefficient of correlation.

18. Define the t, chi-square and F distributions.

SECTION-C

Answer any TWO questions.                                                                           (2×20=40 marks)

19. a) State and prove Lapunov inequality concerning absolute moments.            (10)
20. 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)

1. 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)

1. 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 ³ 2, the binomial distribution can be obtained from the zero-one

distribution.                                                                                                                   (4)

1. b) Examine the additive property for Gamma random variables. (6)
2. c) State and prove Bernoulli’s weak law of large numbers.                      (10)
3. a) State and prove the Chebyshev Show that in the class of random variables

whose second order moment exists, one cannot obtain a better inequality than the

Chebyshev inequality.                                                                                                (10)

1. b) If the ^th moment of a random variable exists, then show that can be expressed

in terms of the ^th derivative of the characteristic function of this random variable

at t = 0.                                                                                                                          (8)

1. c) Define the strong law of large numbers. (2)

Go To Main page

             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2006

                                               ST 3951 – MATHEMATICAL STATISTICS – I

Date & Time : 27-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

     SECTION-A    (10 ´ 2 = 20)

Answer ALL questions.  Each question carries 2 marks.

1. Give an example for a non-decreasing sequence of sets.

1. Distinguish between experiment and random experiment.
2. Let f (x) =    x/15,  x=1,2,3,4,5

0,  otherwise.

Find the median of the above distribution.

1.  Let f(x) = (4-x) / 16, -2 < x < 2 ,zero elsewhere, be the p.d.f. of X.

If Y = ‌ X ‌ , compute P(Y ≤ 1).

5. Give an example of a random variable in which mean doesn’t exist.

1. Prove that E(E(X / Y)) = E(X).
2. Define Hyper Geometric distribution.
3. Define the characteristic function of a multidimensional random vector.

p                         p                                                 p

1.  If    X_n → X    and    Y_n →   Y, then show that X_n + Y_n   →   X +Y.
2. State Lindeberg-Feller theorem.

SECTION-B   (8 x 5 = 40)

Answer any 5 questions.  Each question carries 8 marks.

11. Let f(x) = ½, -1 < x < 1, zero elsewhere, be the p.d.f. of X.  Find the distribution

function and the p.d.f. of Y = X².

12. State and prove Chebyshev’s inequality.
13. If X₁ and X₂ are discrete random variables having the joint p.m.f.

f(x₁,x₂) = ( x₁ + 2 x₂ ) / 18,  (x₁, x_2­) = (1,1), (1,2), (2,1), (2,2), zero elsewhere, determine the conditional mean and variance of X₂, given X₁ =x₁, for x₁ = 1 or 2.

Also, compute E[ 3X₁ – 2 X₂ ].

14. State and prove any two properties of MGF.
15. Stating the conditions, show that binomial distribution tends to Poisson

distribution.

16. Obtain the central moments of N (µ, σ2).
17. Let X ~ G (n₁, α) and Y ~ G (n₂, α) be independent. Find the distribution of X/Y.
18. Explain in detail the difference between WLLN and SLLN.

                                           SECTION-C ( 20 x 2 = 40 )

Answer any 2 questions. Each question carries 20 marks.

19. a) State and prove Bayes’ theorem. (10)
20. b) Bowl I contains 3 red chips and 7 blue chips. Bowl II contains 6 red chips and 4 blue chips. A bowl is selected at random and then 1 chip is drawn from this bowl.  Compute the probability that this chip is red.  Also, relative to the hypothesis that the chip is red, find the conditional probability that is drawn from bowl II.  (10)

20. a) Find the mean and variance of the random variable X having the distribution function:

F(x)   =   0,        x < 0,

( x/4),   0≤x <1,

(x² /4), 1≤x<2,

1 ,        x≥ 2.                                                  (10)

1. b) Let X have the uniform distribution over the interval ( -π/2 , π/2). Find the

distribution of Y  = tan X.    (10)

21. a) State and prove Kolmogorov’s strong law of large numbers. (12)
22. b) State and prove Borel-Cantelli lemma. (8)

22. a) Examine if central limit theorem (using Lyapounov’s condition) holds for the

following sequence of independent variates:

P     X_k   =   ±  2^k      =   2^{-(2k + 1)} ,             P       X_k  =  0    =   1 – 2^–2k   (8)

1. b) State and prove Lindeberg-Levy central limit theorem. (12)

Go To Main page