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              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

 

 Answer ALL the questions:                                                                       [10×2 =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
4. Define improper integral of II kind.
5. Define Gamma integral and state when it converges.
6. Let u = (3y – x) / 6 and v= x / 3. Obtain the Jacobian of transformation.
7. Examine whether f(x,y) = 2, 0 < x < y < 1, is a bivariate probability density function.
8. 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:                                                                      [5×8 =40]

11. Evaluate from first principles.
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

x^m-1(1-x)^n-1dx.

15. Solve the differential equation:

.

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 is the characteristic

root of the matrix .

18. Find the characteristic roots and corresponding vectors of the matrix A where

.

PART – C

 Answer any TWO questions:                                                                     [2×20 =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

.Γ(n+1).

21. (a) Solve the following differential equation using Laplace transform, where y(0) = 3 and

y ¢(0) = 6 :

.

 

(b) Obtain the inverse transform of

 

22. Verify Cayley-Hamilton Theorem for the matrix and hence, find the inverse

of A.

 

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