LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
B.Sc., DEGREE EXAMINATION - STATISTICS
THIRD SEMESTER – APRIL 2004
ST 3500/STA 502 - STATISTICAL MATHEMATICS - II
Max:100 marks
21.04.2004
1.00 - 4.00
SECTION -A
(10  2 = 20 marks)
Answer ALL questions.
1. Define a Skew-Symmetric matrix and give an example.
2. Define an Orthogonal matrix. What can you say about its determinant?
1 1  2
3. Find the rank of 1  1 4  .


3 1
0 
4. State a necessary and sufficient condition for R-integrability of a function.
 1
5. Is 
dx convergent?
x
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.
1
SECTION -B
(5  8 = 40 marks)
Answer any FIVE questions.
11. Define 'upper triangular matrix'.
is an upper triangular matrix.
1 2
12. Find the inverse of A = 0 1
3  1
Show that the product of two upper triangular matrices
1
 1 using Cayley- Hamilton theorem.
1 
Cos3 x
x3
dx b) 
dx
x 1
Sin x
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( X < 1),
P (X < -2)
13. Find a) 
16. Solve:
dy
3x 2 y
Sin 2 x

 
.
dx
1  x3
(1  x 3 )
1
17. Show that the mixed derivative of the following function at the origin are different:
f (x, y) =
xy ( x 2  y 2 )
x2  y 2
, ( x, y )  (0, 0)
0
, ( x, y )  (0, 0)
18. Define Gamma integral and Gamma distribution.
find the mean and variance of the distribution.
SECTION - C
(2  20 = 40 marks)
Answer any TWO questions.
0  1
3  2
0
2
2
1 

19. a) Find the inverse of
using sweep-out process or partitioning
1  2  3  2


1
2
1
0
method.
b) Find the characteristic roots and any characteristic vector associated with them for the
matrix.
4  4
7
 4  8  1
(10+10)


 4  1  8


1
1
dx (iii) 
dx .
2
2 x
2
1

x
x
2 x 1
1
0
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
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)
20. a) Test the convergence of: (i) 
1
dx (ii) 
x
2 x
22. a) Change the order of integration and evaluate:   xy2 dy dx .
1 1
b) Define Beta distributions of I and II kinds.
Find the mean and variance of Beta distribution of I kind
(10+10)

2