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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 B.Sc. DEGREE EXAMINATION – STATISTICS THIRD SEMESTER – APRIL 2008 ST 3501 / 3500 - STATISTICAL MATHEMATICS - II Date : 26/04/2008 Time : 1:00 - 4:00 Dept. No. NO 12 Max. : 100 Marks SECTION - A Answer ALL the Questions (10 x 2 = 20 marks) 1. Define Upper and Lower Sums of a function corresponding to a given partition of a closed interval. 2. State the Second Mean Value theorem for integrals. 3. If f(x) = C x2 (1 – x), for 0 < x < 1 is a probability density function (p.d.f.), find the value of C. 4. Define improper integral of second kind. 1 5. Show that the improper integral x dx converges (where a > 0) a e 1 2 6. Find L–1 ( s a) 3 7. State the general solution for the linear differential equation dy + Py = Q dx 8. State the postulates of a Poisson Process. 9. State the Fundamental Theorem on a necessary and sufficient condition for the consistency of a system of equations A x + b = 0 . ~ ~ ~ 10. If λ is a characteristic root of A, show that λ2 is a characteristic root of A2. SECTION - B Answer any FIVE Questions (5 x 8 = 40 marks) 11. Let Pn = {0, 1/n, 2/n, ….., (n – 1)/n, 1} be a partition of [0, 1]. For the function f(x) = x, 0 ≤x ≤ 1, find lim U[Pn, f] and lim L[Pn, f]. Comment on the integrability of the function. n n 1 12. Evaluate: (i) x 0 1 x /4 d x (ii) 4 x dx 0 13. Show that the integral Sin 1 x p dx (where a > 0) converges for p > 1 and diverges for p ≤ 1. a 14. Define Beta integrals of First kind and Second kind. Show that one can be obtained from the other by a suitable transformation. (P.T.O) 2 3 dy 3x y y 15. Solve: = dx 3 y 2 x x3 16. Solve: (D2 + 4D + 6) y = 5 e– 2 x 1 17. Establish the relationship between the characteristic roots and the trace and determinant of a matrix. 18. Give a parametric form of solution to the following system of equations: 4x1 – x2 + 6x3 = 0 2x1 + 7x2 +12x3 = 0 x1 – 4x2 – 3x3 = 0 5x1 – 5x2 +3x3 = 0 SECTION - C Answer any TWO Questions (2 x 20 = 40 marks) 19. (a) State and prove the First Fundamental Theorem of Integral Calculus. 2 1 5x2 x4 (b) Evaluate (i) 2 dx (ii) 2 dx 0 x 4 0 x 1 (12+8) 20. (a) Show that mean does not exist for the distribution with p.d.f. 1 1 f(x) = ,–∞<x<∞ 1 x 2 d (b)L[f(t)] = F(s), show that L[t f(t)] = – F(s). Using this result find L[t2 e– 3 t] ds (8+12) 21. Evaluate: (a) ∫ ∫ x2 y2 dx dy over the circle x2 + y2 ≤ 1. (b) ∫ ∫ y dx dy over the region between the parabola y = x2 and the line x + y = 2 (10+10) 22. (a) State and Prove Cayley-Hamilton Theorem. (b) If P is a non-singular matrix and A is any square matrix, show that A and P–1AP have the same characteristic equation. Also, show that if ‘x’ is a characteristic vector of A, then P–1 x is a characteristic vector of P–1AP. (12+8) ******* 2