LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034 B.Sc., DEGREE EXAMINATION - STATISTICS THIRD SEMESTER – APRIL 2004 ST 3500/STA 502 - STATISTICAL MATHEMATICS - II Max:100 marks 21.04.2004 1.00 - 4.00 SECTION -A (10 2 = 20 marks) Answer ALL questions. 1. Define a Skew-Symmetric matrix and give an example. 2. Define an Orthogonal matrix. What can you say about its determinant? 1 1 2 3. Find the rank of 1 1 4 . 3 1 0 4. State a necessary and sufficient condition for R-integrability of a function. 1 5. Is dx convergent? x 6. If f(x) = C x2, 0 < x < 1, is a probability density function (p.d.f), find 'C'. 7. Give an example of a homogeneous differential equation of first order. 8. Distinguish between 'double' and 'repeated' limits. 9. State any two properties of a Bivariate distribution function. 10. State the rule of differentiation of a composite function of two variables. 1 SECTION -B (5 8 = 40 marks) Answer any FIVE questions. 11. Define 'upper triangular matrix'. is an upper triangular matrix. 1 2 12. Find the inverse of A = 0 1 3 1 Show that the product of two upper triangular matrices 1 1 using Cayley- Hamilton theorem. 1 Cos3 x x3 dx b) dx x 1 Sin x 14. State and prove first Fundamental Theorem of Integral Calculus. 15. If X has p.d.f f(x) = x2/18, -3 x 3, find the c.d.f of X. Also, find P( X < 1), P (X < -2) 13. Find a) 16. Solve: dy 3x 2 y Sin 2 x . dx 1 x3 (1 x 3 ) 1 17. Show that the mixed derivative of the following function at the origin are different: f (x, y) = xy ( x 2 y 2 ) x2 y 2 , ( x, y ) (0, 0) 0 , ( x, y ) (0, 0) 18. Define Gamma integral and Gamma distribution. find the mean and variance of the distribution. SECTION - C (2 20 = 40 marks) Answer any TWO questions. 0 1 3 2 0 2 2 1 19. a) Find the inverse of using sweep-out process or partitioning 1 2 3 2 1 2 1 0 method. b) Find the characteristic roots and any characteristic vector associated with them for the matrix. 4 4 7 4 8 1 (10+10) 4 1 8 1 1 dx (iii) dx . 2 2 x 2 1 x x 2 x 1 1 0 b) Define Lower and Upper sum in the context of Riemann integration. Show that lower sums increase as partitions become finer. (12+8) 21. a) Investigate the maximum and minimum of f(x,y) = 21x - 12x2 - 2y2 + x3 + xy2 b) If f(x,y) = e-x-y, x,y > 0, is the p.d.f of (X, Y), find the distribution function. (12+8) 20. a) Test the convergence of: (i) 1 dx (ii) x 2 x 22. a) Change the order of integration and evaluate: xy2 dy dx . 1 1 b) Define Beta distributions of I and II kinds. Find the mean and variance of Beta distribution of I kind (10+10) 2