LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc.

AC 06

DEGREE EXAMINATION –STATISTICS

THIRD SEMESTER – APRIL 2007

ST 3500 – STATISTICAL MATHEMATICS – II

 

 

Date & Time: 21/04/2007 / 1:00 – 4:00            Dept. No.                                                     Max. : 100 Marks

 

 

SECTION A

 

Answer ALL questions. Each carries 2  marks                                 [10×2=20]

 

1. Define Skew-Hermitian matrix and give an example of 3×3 skew Hermitian

matrix.

2. State Cayley-Hamilton theorem.
3. Evaluate the primitive integral : .
4. Define order and degree of differential equations and give an example
5. What is meant by double limit? Give an example
6. Define improper integral of second kind
7. Define Integrability and Integral of a function
8. Solve (2- 4x²)dy = (6x-xy) dx
9. If f(x) =     is a p.d.f. , find the value of K
10. Evaluate

 

SECTION B

Answer any FIVE questions                                                                    (5×8 =40)

11. Find the inverse of the matrix A =by using 2×2 partitioning
12. State and prove a necessary and sufficient condition for integrability of a

function

13. Evaluate :
14. Define Beta distribution of 1^st kind and hence find its mean and variance by stating the conditions for their existence.

 

15. Show that double limit at the origin may not exist but repeated limits exist for

the following function :

f(x,y) =

 

 

16. Investigate the extreme values of f(x,y) = (y-x)⁴ + (x-2)², x, y Î R.

 

17. Prove that

 

 

 

18. Compute mean and variance for the following p.d.f

 

SECTION C

Answer any TWO questions                                                             (2 x 20 =40)        

19.a] Find the rank of the matrix A=

 

[b] Find the characteristic roots of the following matrix. Also find the

inverse using  Cayley-Hamilton theorem:

A=

20. a] Test if    converges absolutely

b] Solve the differential equation

21. a] If f(x,y) =

is the joint p.d.f of (X,Y), find the joint d.f. F(x,y).

[b] Let  , x, y Î R. Find  f_xx , f_yy , f_xy f_yx

22. [a] Let f(x,y) = be the joint p.d.f of (x,y). Find the

co-efficient of  correlation    between X and Y.

[b]  Change the order integration and evaluate

 

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