LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2012

ST 2502/ST 2501/ST 2500 – STATISTICAL MATHEMATICS – I

 

Date : 16-04-2012                   Dept. No.                                          Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL the Questions:                                                                                          (10 x 2 = 20 marks)

1. Define monotonically decreasing sequences.
2. Define random variable.
3. Define divergence sequences.
4. What is meant by linear dependence?
5. 5. Find the trace of the matrix A = 
6. State Rolle’s Theorem.
7. The probability distribution of a random variable X is: Determine       the constant k.
8. Define symmetric matrix. Give an example.
9. Find the determinant of the matrix
10. Define stochastic matrix.

PART – B

Answer any FIVE questions:                                                                                                    (5 x 8 = 40 marks)

11. The diameter, say X, of an electric cable, is assumed to be continuous random variable with p.d.f
12. i) Check that the above is a p.d.f. ;   ii) Obtain an expression for the c.d.f of x ;

iii)  Compute ;  iv) Determine the number K such that P(X < k) = P(X > k)

12. Prove that a convergent sequence is also bounded.
13. By using first principles, show that the sequences , where, n = 1, 2, . . . ,

converges to   .

14. Show that differentiability of a function at a point implies continuity. What can you say about the

converse? Justify your answer.

15. State and prove Lagrange’s Mean Value Theorem. (P.T.O.)
16. Obtain the Maclaurin’s Series expansion for log(1+x), for – 1 < x < 1 .
17. If the joint distribution function of X and Y is given by
18. a) Find the marginal densities of X and of Y ;     b) Are X and Y independent?
19. c) Find P(X  1 Y ;
20. Find inverse of the matrix

PART – C

Answer any TWO questions:                                                                                                (2 x 20 = 40 marks)

19. Examine the validity of the hypothesis and the conclusion of Rolle’s theorem for the          function f defined in  in each of the following cases:
20. i) , a = 0, b = 2
21. ii) , a = -3, b = 0
22. Two fair dice are thrown. Let X₁ be the score on the first die and X₂ the score on the second die. Let Y denote the maximum of X₁ and X₂ i.e. max(X₁, X₂).
23. a) Write down the joint distribution of Y and X₁.
24. b) Find E (Y), Var (y) and Cov (Y, X₁).
25. Suppose that two-dimensional continuous random variable (X, Y) has joint probability density        function given by
26.  i) Verify that
27.   ii) Find P (0 < X <,  P(X+Y < 1),  P(X > Y),  P(X < 1 | Y < 2)
28. (a) Find the rank of .

(b) Verify whether the vectors (2, 5, 3), (1, 1, 1) and (4,–2, 0) are linearly independent.       (10 +10)

 

 

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