LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2003

ST-3500/STA502 – STATISTICAL MATHEMATICS – II

04.11.2003                                                                                                           Max:100 marks

9.00 – 12.00

SECTION-A

Answer ALL the questions.                                                                              (10×2=20 marks)

 

1. If P^* is a partition of [a , b] finer than the partition P, state the inequality governing the upper sums lower sums of a function f corresponding to P and P^*.
2. Find .
3. State the first Fundamental Theorem of Integral calculus.
4. Solve: .
5. “The function f(x,y) =   xy/(x²+y²) ,    (x,y)  ¹(0,0)

 

0                ,   (x, y) = (0, 0)

does not have double limit as (x, y)   – verify.

6. State the rule for the partial derivative of a composite function of two variables.
7. Define Gamma distribution.
8. Write down the Beta integral with integrand involving Sine and Cosine functions.
9. Define a symmetric matrix.
10. Find the rank of the matrix .

 

SECTION-B

Answer any FIVE  questions.                                                                          (5×8=40 marks)

 

11. Evaluate (a) . (4+4)

(b)

12. If f(x) = kx² , 0 < x¸< 2 , is the probability density function (p.d.f) of X, find (i) k

(ii) P[X<1/4],  (iii) P,  (iv) P[X >1].

13. Solve the non-homogeneous differential equation:

(y – x – 3) dy = (2x + y +6) dx

14. For the function          xy(x² – y²) / (x² + y²)  ,    (x,y)  ¹(0,0)

f(x,y) =

0                                ,      (x, y) = (0, 0)

Show that f_x (x, 0) = f_y (0,y) = 0 , f_x (0, y) = -y , f_y (x, 0) = x.

15. Find the mean and variance of Beta distribution of II kind stating the conditions for their existence.
16. If f(x,y) = e^-x-y , x > 0, y > 0, is the joint p.d.f of (x, y),  find the joint c.d.f. of (x, y).  Verify that the second order mixed derivative of the joint c.d.f.is indeed the joint p.d.f.
17. Establish the reversal law for Transpose of product of matrices. Show that the operations of Inversion and Transpositions are commutative.
18. Find the inverse of using Cayley – Hamilton Theorem.

 

SECTION-C

Answer any TWO  questions.                                                                          (2×20=40 marks)

 

19. a) Show that, if fÎ R [a, b] then f² Î R [ a, b].
20. b) If f(x) = c.e^-x, x > 0, is the p.d.f. of X, find (i) c (ii) E(X), (iii) Var (X).
21. c) Discuss the convergence of (8+6+6)
22. a) Investigate for extreme values of the function

f (x, y) = x³ + y³ – 12x – 3y + 5, x, y Î R.

1. b) Define joint distribution function for bivariate case and state its properties. Establish

the property which gives the probability P[x₁ < X £ x_2,  y₁ < Y £ y₂] in terms of the

joint distribution function of (X, Y).                                                               (10+10)

21. If x + y ,   0 < x, y < 1

f (x, y) =

0        ,   otherwise

is the joint p.d.f of (x, y),  find the means and variances  of X and Y and covariance

between X and Y.  Also find  P [ Y < X] and the marginal p.d.f’s of X and Y.

22. a) By partitioning into 2 x 2 submatrices find the inverse of
23. b) Find the characteristic roots and any characteristic vectors for the matrix

 

(10 + 10)

 

 

Go To Main page

Latest Govt Job & Exam Updates:

View Full List ...