LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION –STATISTICS
AC 06
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
[10x2=20]
1. Define Skew-Hermitian matrix and give an example of 3x3 skew Hermitian
matrix.
2. State Cayley-Hamilton theorem.
e4x
 e 4 x  5dx .
Define order and degree of differential equations and give an example
What is meant by double limit? Give an example
Define improper integral of second kind
Define Integrability and Integral of a function
Solve (2- 4x2)dy = (6x-xy) dx
3. Evaluate the primitive integral :
4.
5.
6.
7.
8.
 K( x  3), 1  x  2
9. If f(x) = 
0, otherwise

10. Evaluate

2
0

is a p.d.f. , find the value of K
sin 7  cos7 d
SECTION B
Answer any FIVE questions
(5x8 =40)
1  2  3  2 
0 2
2
1 
 by using 2x2 partitioning
11. Find the inverse of the matrix A = 
3  2 0  1


2
1 
0 1
12. State and prove a necessary and sufficient condition for integrability of a
function
2x  5
dx
13. Evaluate :  2
x  10 x  2
14. Define Beta distribution of 1st 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 :
xy

, (x, y)  (0,0)
 2
f(x,y) =  x  y 2

(x, y)  (0,0)
0,
16. Investigate the extreme values of f(x,y) = (y-x)4 + (x-2)2, x, y  R.
17. Prove that lim( x 2  2y)  5 as ( x, y)  (1,2)
18. Compute mean and variance for the following p.d.f
 6x (1  x ), if 0  x  1
f (x)  
otherwise
0,
SECTION C
Answer any TWO questions
(2 x 20 =40)
1
 4

19.a] Find the rank of the matrix A=  2

3
 4
2 0
2
1
1 1
4
2 
5 1
1
0

8  1 3  1
3 0  2 4 
[b] Find the characteristic roots of the following matrix. Also find the
inverse using Cayley-Hamilton theorem:
3
1 0

A= 2 1  1


1  1 1 
20. a] Test if


e  x sin xdx converges absolutely

b] Solve the differential equation
dy y  5

dx x  6
e  x  y , x  0, y  0
21. a] If f(x,y) = 
 0, otherwise)
is the joint p.d.f of (X,Y), find the joint d.f. F(x,y).
2x  1
[b] Let f ( x, y) 
, x, y  R. Find fxx , fyy , fxy fyx
xy
 x  y, 0  x  1,0  y  1
22. [a] Let f(x,y) = 
be the joint p.d.f of (x,y). Find the
0, otherwise)

co-efficient of correlation between X and Y.
2
[b] Change the order integration and evaluate

1
x
1
xy 2dydx
******
2