LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AB 08

THIRD SEMESTER – NOV 2006

ST 3500 – STATISTICAL MATHEMATICS – II

(Also equivalent to STA 502)

 

 

Date & Time : 02-11-2006/9.00-12.00     Dept. No.                                                   Max. : 100 Marks

 

 

 

SECTION A

Answer ALL questions. Each carries 2  marks                                 [10×2=20]

 

1. Define Hermitian matrix and give an example of 3×3 Hermitian matrix
2. Write the formula for finding the determinant by using partioning of matrices
3. Define Riemann integral
4. Evaluate
5. Define repeated limits and give an example
6. Define improper integral of I^st   kind
7. What are the order and degree of the differential equation ?
8. Define continuity of functions of two variables
9. Evaluate
10. If X, Y are random variables with joint distribution  function F(x,y), express          Pr[k₁< X ≤ k_{2 ,}   m₁ < Y ≤ m₂] in terms of

 

 

SECTION B

Answer any FIVE questions                                                                    (5×8 =40)

 

11. State and prove the first fundamental theorem of integral calculus
12. Find the inverse of the matrix A =by using 2×2 partitioning
13. Discuss the convergence of following improper integrals:

[a]     [b]

14. Define Gamma distribution and hence derive its mean and variance
15. Solve the differential eqation

 

 

 

 

16. Investigate the existence of the repeated limits and double limit at the origin of the

function  f(x,y) =

17. Investigate for extreme values of f(x,y) = (y-x)⁴ + (x-2)², x, y Î R.

 

18 . Define Beta distribution of 2^nd  kind.  Find its mean and variance by stating the

conditions for their existence.

SECTION C

Answer any TWO questions                                                             (2 x 20 =40)        

19. [a] Find the rank of the matrix A=

[b] Find the characteristic roots of the following matrix. Also find the inverse of A

using  Cayley-Hamilton theorem, where

A=

20. [a] Test if    converges absolutely

[b] Compute mean, mode and variance for the following p.d.f

 

21. a] Find maximum or minimum of f(x,y) = (x-y)² + 2x – 4xy, x, y Î

b] Show that the mixed derivatives of the following function at the origin are

different:

22.a] Let f(x,y) =    be  the joint p.d.f of (x,y).

Find the co-efficient of correlation between X and Y.

[b]  Change the order of integration and evaluate

 

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