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.
0
