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)

 

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