LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
THIRD SEMESTER – APRIL 2012
ST 3503/ST 3501/ST 3500 - STATISTICAL MATHEMATICS - II
Date : 24-04-2012
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
PART – A
Answer ALL the questions:
[10x2 =20]
1. Define upper sum and lower sum of a function defined over the
interval [a, b].
2. Examine whether the function f(x) = 1/x2 , for x≥1, is a p.d.f. If so find
EX.
3. Define improper integral of II kind.
4. Define Gamma integral and state when it converges.
5. Let u = (3y – x) / 6 and v= x / 3. Obtain the Jacobian of transformation.
6. Examine whether f(x,y) = 2, 0 < x < y < 1, is a bivariate probability density function.
7. When do we say that a differential equation is variables separable? Show that (1 – x) dy –
(3 + y) dx = 0 is variables separable.
8. Obtain the Laplace transform of g(t) = e-λt , t > 0.
9. Define characteristic equation and characteristic roots.
10. State Cayley Hamilton Theorem.
PART – B
Answer any FIVE questions:
[5x8 =40]
1
11. Evaluate
 xdx
from first principles.
0
12. Show that every continuous function defined on a closed interval of the real line is
Riemann integrable.
13. Obtain the moment generating function of the two parameter Gamma distribution. Hence
find the mean and variance.
14. Discuss the convergence of the integral
1

0
xm-1(1-x)n-1dx.
15. Solve the differential equation:
d 2 y dy

 2 y  x  sin x .
dx 2 dx
16. State and prove Initial Value and Final Value Theorems of Laplace transforms.
17. If λ is the characteristic root of a non-singular matrix A, show that  1 is the characteristic
root of the matrix A 1 .
18. Find the characteristic roots and corresponding vectors of the matrix A where
1 2 3 
A  0 2 3 .
0 0 2
PART – C
Answer any TWO questions:
[2x20 =40]
19. (a) If f(x) and g(x) are two Riemann-integrable functions, then show that the sum
f(x) + g(x) is also Riemann integrable.
(b) State and Prove the First Fundamental Theorem of integral calculus.
20. (a) Establish the relation between the Beta function and Gamma function. Hence find the
value of β(1/2, 1/2).
(b) For a non-negative function f(t) =tn show that
L f (t ) 
1
s n 1
.Γ(n+1).
21. (a) Solve the following differential equation using Laplace transform, where y(0) = 3 and
y (0) = 6 :
d2y
dy
2
8 y  0 .
2
dt
dt
(b) Obtain the inverse transform of
2  4s
s  4 s  13
2
1 2 1
22. Verify Cayley-Hamilton Theorem for the matrix A  0 1 - 1 and hence, find the inverse
3 - 1 1
of A.
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