LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – APRIL 2004

ST 3500/STA 502 – STATISTICAL MATHEMATICS – II

21.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

 

SECTION -A

 

Answer ALL questions.                                                                              (10 ´ 2 = 20 marks)

 

1. Define a Skew-Symmetric matrix and give an example.
2. Define an Orthogonal matrix. What can you say about its determinant?
3. Find the rank of .
4. State a necessary and sufficient condition for R-integrability of a function.
5. Is convergent?
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.

 

SECTION -B

 

Answer any FIVE questions.                                                                                  (5 ´ 8 = 40 marks)

 

11. Define ‘upper triangular matrix’. Show that the product of two upper triangular matrices is an upper triangular matrix.
12. Find the inverse of A = using Cayley- Hamilton theorem.
13. Find a)     b)
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(< 1),

P (X < -2)

16. Solve: .

 

 

 

17. Show that the mixed derivative of the following function at the origin are different:

 

 

f (x, y) =

 

 

18. Define Gamma integral and Gamma distribution.

find the mean and variance of the distribution.

 

SECTION – C

 

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

 

19. a) Find the inverse of using sweep-out process or partitioning

method.

1. b) Find the characteristic roots and any characteristic vector associated with them for the

matrix.

(10+10)

20. a) Test the convergence of: (i) (ii)   (iii) .
21. 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

1. 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)
2. a) Change the order of integration and evaluate: .
3. b) Define Beta distributions of I and II kinds.

Find the mean and variance of Beta distribution of I kind                                  (10+10)

 

 

Go To Main page

Latest Govt Job & Exam Updates:

View Full List ...