LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
SECOND SEMESTER – APRIL 2012
ST 2502/ST 2501/ST 2500 - STATISTICAL MATHEMATICS - I
Date : 16-04-2012
Dept. No.
Max. : 100 Marks
Time : 9:00 - 12:00
PART – A
Answer ALL the Questions:
(10 x 2 = 20 marks)
1. Define monotonically decreasing sequences.
2. Define random variable.
3. Define divergence sequences.
4. What is meant by linear dependence?
𝟏 𝟑
5. Find the trace of the matrix A = [𝟐 −𝟏
𝟑 𝟏
𝟒
𝟔]
𝟒
6. State Rolle’s Theorem.
1
7. The probability distribution of a random variable X is: 𝑓(𝑥) = 𝑘 sin 5 𝜋𝑥, 0 ≤ 𝑥 ≤ 5. Determine
the constant k.
8. Define symmetric matrix. Give an example.
9. Find the determinant of the matrix 𝐴 = [
5 1
]
2 6
10. Define stochastic matrix.
PART - B
Answer any FIVE questions:
(5 x 8 = 40 marks)
11. The diameter, say X, of an electric cable, is assumed to be continuous random variable with p.d.f
𝑓(𝑥) = 6𝑥 (1 − 𝑥), 0 ≤ 𝑥 ≤ 1
i) Check that the above is a p.d.f. ; ii) Obtain an expression for the c.d.f of x ;
iii) Compute 𝑃 (𝑋 ≤
1
2
|
1
3
2
≤ 𝑋 ≤ 3) ; iv) Determine the number K such that P(X < k) = P(X > k)
12. Prove that a convergent sequence is also bounded.
13. By using first principles, show that the sequences < 𝑓𝑛 ≥ , where𝑓𝑛 =
converges to
1
2
𝑛2 + 1
2𝑛2 + 5
, n = 1, 2, . . . ,
.
14. Show that differentiability of a function at a point implies continuity. What can you say about the
converse? Justify your answer.
15. State and prove Lagrange’s Mean Value Theorem.
(P.T.O.)
16. Obtain the Maclaurin’s Series expansion for log(1+x), for – 1 < x < 1 .
17. If the joint distribution function of X and Y is given by
1 − 𝑒 −𝑥 − 𝑒 −𝑦 + 𝑒 − (𝑥+𝑦) ; 𝑥 > 0, 𝑦 > 0
𝐹𝑋𝑦 (𝑥, 𝑦) = {
0,
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
a) Find the marginal densities of X and of Y ;
c) Find P(X ≤ 1 ∩ Y ≤ 1);
b) Are X and Y independent?
0 1 1
18. Find inverse of the matrix [1 0 1]
1 1 0
PART - C
Answer any TWO questions:
19.
(2 x 20 = 40 marks)
Examine the validity of the hypothesis and the conclusion of Rolle’s theorem for the
function f defined in [𝑎, 𝑏], in each of the following cases:
2
i) 𝑓(𝑥) = 2 + (𝑥 − 1)3 , a = 0, b = 2
𝑥
ii) 𝑓(𝑥) = 𝑥 (𝑥 + 3)𝑒 −2 , a = -3, b = 0
20. Two fair dice are thrown. Let X1 be the score on the first die and X2 the score on the second die.
Let Y denote the maximum of X1 and X2 i.e. max(X1, X2).
a) Write down the joint distribution of Y and X1.
b) Find E (Y), Var (y) and Cov (Y, X1).
21. Suppose that two-dimensional continuous random variable (X, Y) has joint probability density
function given by
2
𝑓(𝑥, 𝑦) = {6 𝑥 𝑦, 0 < 𝑥 < 1, 0 < 𝑦 < 1
0,
𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
1
1
i) Verify that ∫0 ∫0 𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦 = 1
3
ii) Find P (0 < X <4 ,
1
3
< 𝑌 < 2), P(X+Y < 1), P(X > Y), P(X < 1 | Y < 2)
1 2
22. (a) Find the rank of [2 −2
3 0
3 −2
1 3 ].
4 1
(b) Verify whether the vectors (2, 5, 3), (1, 1, 1) and (4,–2, 0) are linearly independent.
$$$$$$
(10 +10)