# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 ```LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
THIRD SEMESTER – NOVEMBER 2012
ST 3503/3501/3500 - STATISTICAL MATHEMATICS - II
Date : 02/11/2012
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
PART-A
[10x2 =20]
1. Define upper sum and lower sums.
2. Give an example of a function that is not Riemann integrable.
3. Define improper integrals.
4. State the comparison test for the improper integrals
5. Define variance-covariance matrix.
6. When do we say that a system of equations is homogenous?
7. State the order and degree of the differential equation:
d2y
dy
 5( x  y )  3xy  x 2
2
dx
dx
3t
5 t
8. Obtain the Laplace transform of f (t )  e  e  , t &gt; 0.
x
9. Define characteristic equation and characteristic roots.
10. Write down the importance of Caley-Hamilton Theorem.
PART – B
[5x8 =40]
1
11. Evaluate
 x dx
2
from first principles.
0
12. Show that every continuous function defined on a closed interval of the real line is
Riemann integrable.
13. Find the mean and variance of Beta distribution of I kind.
14. Discuss the convergence of gamma integral.
2, 0  x  y 1
.
 0, otherwise
15. Find the covariance between X and Y whose joint p.d.f. is f ( x, y )  
16. Evaluate
  x
2
 y 2 dx dy over the upper half of the circle with centre (0, 0) and radius 1.
17. a) Solve the differential equation
d2y
dy
 3  3y  e4x .
2
dx
dx
(b) Obtain the inverse Laplace transform of
2  4s
.
s  4s  3
2
18.
Find
the
characteristic
roots
and
corresponding
vectors
of
the
matrix
1 2 3 
A  0 2 3 .
0 0 2
PART - C
[2x20 =40]
19. (a) State and Prove the first fundamental theorem on integral calculus.
(b) Derive the MGF of normal distribution. Hence find its mean and variance.
20. (a) Establish the relation between the Beta and Gamma integrals. Hence find the
value of β (3, 4).
(b) Solve the differential equation:
d2y
dy
 5  6 y  e4x .
2
dx
dx
21. (a) Solve the following initial value problem using Laplace transform,
where y(0) = -2 and y'(0) = 5 :
d 2 y dy

 2y .
dt 2 dt
(b) Let X and Y be two independent one parameter Gamma random variables with
parameters 1 and 2 respectively. Use ‘transformation of variables’ method to
obtain the distribution of
X
.
X Y
22. (a) Show that if λ is a characteristic root of A, then λn is a characteristic root of An.
(b) Solve the system of equations using matrix inverse method.
x  y  z 1
x yz 3
x  2 y  3z  0 .
\$\$\$\$\$\$\$
```