04.11.2003
9.00 - 12.00
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
B.Sc., DEGREE EXAMINATION - STATISTICS
THIRD SEMESTER – NOVEMBER 2003
ST-3500/STA502 - STATISTICAL MATHEMATICS - II
Max:100 marks
SECTION-A
Answer ALL the questions.
(10x2=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*.
2x
dx .
2. Find  2
( x  1) ( x 2  3)
3. State the first Fundamental Theorem of Integral calculus.
4. Solve: e x tan y dx  (1 e x ) sec 2 y dy  0 .
5. "The function f(x,y) = xy/(x2+y2) ,
(x,y) (0,0)
0
, (x, y) = (0, 0)
does not have double limit as (x, y)  (0, 0)". - verify.
6.
7.
8.
9.
State the rule for the partial derivative of a composite function of two variables.
Define Gamma distribution.
Write down the Beta integral with integrand involving Sine and Cosine functions.
Define a symmetric matrix.
  1 3  2
10. Find the rank of the matrix  2
1  1 .
 3  2 1 
SECTION-B
Answer any FIVE questions.
0
e
11. Evaluate (a)

x
1
dx .
 ex
(5x8=40 marks)
(4+4)
3
(b)
x 24
2 x10  1 dx
12. If f(x) = kx2 , 0 < x¸< 2 , is the probability density function (p.d.f) of X, find (i) k
1

(ii) P[X<1/4], (iii) P   X  2  , (iv) P[X >1].
2

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) (0,0)
f(x,y) =
0
,
(x, y) = (0, 0)
Show that fx (x, 0) = fy (0,y) = 0 , fx (0, y) = -y , fy (x, 0) = x.
1
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.
3
1 0

18. Find the inverse of 2 1  1 using Cayley - Hamilton Theorem.


1  1 1 
SECTION-C
Answer any TWO questions.
(2x20=40 marks)
19. a) Show that, if f R [a, b] then f2  R [ a, b].
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).

c) Discuss the convergence of
x
0
1
2
 x
dx
(8+6+6)
20. a) Investigate for extreme values of the function
f (x, y) = x3 + y3 - 12x - 3y + 5, x, y  R.
b) Define joint distribution function for bivariate case and state its properties. Establish
the property which gives the probability P[x1 < X  x2, 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
3  2 0  1 
0 2
2
1 

1  2  3  2


2
1 
0 1
b) Find the characteristic roots and any characteristic vectors for the matrix
4  4
7
 4  8  1


 4  1  8
(10 + 10)

2