LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
SECOND SEMESTER – APRIL 2011
ST 2502/ST 2501 - STATISTICAL MATHEMATICS - I
Date : 08-04-2011
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
PART – A
Answer ALL questions
(10x2 =20 Marks)
1. Define least upper bound of a set.
2. Define convergent sequence.
3. Define cumulative distribution function and state any two of its properties.
4. Give an example for a monotonic sequence.
5. Define absolute convergence and conditional convergence for a series of real numbers.
6. Define M.G. F of a random variable.
7. State Roll’s theorem.
8. Define Taylor’s expansion of a function about x = a.
9. Define rank of a matrix.
10. Define symmetric matrix. Give an example.
PART – B
Answer any FIVE questions
(5x8=40 Marks)
11. Show that every convergent sequence is bounded. Is the converse true? Justify your answer.
12. Obtain the c.d.f. of the total number of heads occurring in three tosses of a fair coin.

13. Establish the convergence of (a)
1
; (b)

2
n 1 n

1
 (2(n  1))! .
n 1
14. Show that if a function is derivable at a point, then it is continuous at that point.
15. If two random variables X and Y have the joint probability density
2
 ( x  2 y ) ; 0  x  1, 0  y  1
f ( x, y )   3
 0 ; otherwise
Find the marginal densities.
16. Find the Lagrange’s and Cauchy’s remainder after nth term in the Taylor’s series expansion
of loge(1+ x).
17. Verify whether or not the following sets of vectors form linearly independent sets:
(a) (1, 2, 3), (2, 2, 0)
(b) (3, 1, -4), (2, 2, -3)
1 2 3 
18. Find the inverse of a matrix 3 2 3 .
1 1 2
PART – C
Answer any TWO questions
(2x20=40 Marks)
19. (a) Prove that a non-increasing sequence of real numbers which is bounded below is
convergent.
(b)Prove that the sequence a n  given by a n  1 
1 1
1
  ...  is convergent.
1! 2!
n!
20. (a) State and Prove Rolle’s Theorem
(b) Find a suitable c of Rolle’s Theorem for the function
f ( x)  ( x  2) ( x  4) for 2  x  4 .
21. A random variable X has the following probability function
x
1
2
3
4
5
6
7
P(x)
0
k
2k
2k
k2
2k2
7k2+k
(i) Find k
(ii) Evaluate (a) P( X  6) (b) P( X  6) (c) P(0  X  5)
1
(iii) If P( X  k )  , find the minimum value of k
2
(iv) Determine the distribution function of X.
(6  x  y) /8 ; 0  x  2 , 2  y  4
other wise
;
0
22. (a) If f ( x)  
is the joint p.d.f. of X and Y, find the marginal p.d.f.’s. Also, evaluate
P[ (X < 1)
(Y < 3) ]
1 2 3 4 
(b) Find the rank of the matrix A  2 4 6 8  .
3 6 9 12
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