LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
THIRD SEMESTER – NOV 2006
AB 08
ST 3500 - STATISTICAL MATHEMATICS - II
(Also equivalent to STA 502)
Date & Time : 02-11-2006/9.00-12.00
Dept. No.
SECTION A
Answer ALL questions. Each carries 2 marks
Max. : 100 Marks
[10x2=20]
1. Define Hermitian matrix and give an example of 3x3 Hermitian matrix
2. Write the formula for finding the determinant by using partioning of matrices
3. Define Riemann integral
1
dx
4. Evaluate  2
x  6x  5
5. Define repeated limits and give an example
6. Define improper integral of Ist kind
3
d 3y
 dy 
 10x 2    4 y  0 ?
7. What are the order and degree of the differential equation
3
dx
 dx 
8. Define continuity of functions of two variables
9. Evaluate


2
0
sin 9  cos 9  d
10. If X, Y are random variables with joint distribution
Pr[k1< X ≤ k2 , m1 < Y ≤ m2] in terms of F.
function F(x,y), express
SECTION B
Answer any FIVE questions
(5x8 =40)
11. State and prove the first fundamental theorem of integral calculus
3  1 0  1 
0 1
1
1 
12. Find the inverse of the matrix A = 
by using 2x2 partitioning
1  1  3  2


4
2
0 1
13. Discuss the convergence of following improper integrals:


dx
dx
[a] 
[b]  2
1
1
x  x
x(1  x )
14. Define Gamma distribution and hence derive its mean and variance
dy
dy
 xy
15. Solve the differential eqation y 2  x 2
dx
dx
16. Investigate the existence of the repeated limits and double limit at the origin of the
 x. sin( 1 / y )  y. sin( 1 / x ), (x, y)  (0,0)
function f(x,y) = 
0,
(x, y)  (0,0)

17. Investigate for extreme values of f(x,y) = (y-x)4 + (x-2)2, x, y  R.
18 . Define Beta distribution of 2nd kind. Find its mean and variance by stating the
conditions for their existence.
SECTION C
Answer any TWO questions
(2 x 20 =40)
5  2 0  1 3 
0 2
2
1  2


19. [a] Find the rank of the matrix A= 1  2  3  2 1 


2
1  6
0 1
0 2
2
1  2
[b] Find the characteristic roots of the following matrix. Also find the inverse of A
using Cayley-Hamilton theorem, where
 1 1 2 
A=  1 2  1


 2  1 1 
20. [a] Test if


e  x sin xdx converges absolutely
2

[b] Compute mean, mode and variance for the following p.d.f
 6x (1  x ), if 0  x  1
f (x)  
otherwise
0,
21. a] Find maximum or minimum of f(x,y) = (x-y)2 + 2x – 4xy, x, y  R.
b] Show that the mixed derivatives of the following function at the origin are
 4xy ( x 2  y 2 )

, ( x, y)  (0,0)
different:
g( x , y )   x 2  y 2

0, ( x, y)  (0,0)

2, 0  x  y  1
22.a] Let f(x,y) = 
be the joint p.d.f of (x,y).
 0, otherwise
Find the co-efficient of correlation between X and Y.

[b] Change the order of integration and evaluate

0

e y
x y
_________________________________
dydx.