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 Dept. No. Max. : 100 Marks
Time : 9:00 – 12:00
PART – A
Answer ALL questions (10×2 =20 Marks)
- Define least upper bound of a set.
- Define convergent sequence.
- Define cumulative distribution function and state any two of its properties.
- Give an example for a monotonic sequence.
- Define absolute convergence and conditional convergence for a series of real numbers.
- Define M.G. F of a random variable.
- State Roll’s theorem.
- Define Taylor’s expansion of a function about x = a.
- Define rank of a matrix.
- Define symmetric matrix. Give an example.
PART – B
Answer any FIVE questions (5×8=40 Marks)
- Show that every convergent sequence is bounded. Is the converse true? Justify your answer.
- Obtain the c.d.f. of the total number of heads occurring in three tosses of a fair coin.
- Establish the convergence of (a) ; (b) .
- Show that if a function is derivable at a point, then it is continuous at that point.
- If two random variables X and Y have the joint probability density
Find the marginal densities.
- Find the Lagrange’s and Cauchy’s remainder after nth term in the Taylor’s series expansion of loge(1+ x).
- 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)
- Find the inverse of a matrix .
PART – C
Answer any TWO questions (2×20=40 Marks)
- (a) Prove that a non-increasing sequence of real numbers which is bounded below is convergent.
(b)Prove that the sequence given by is convergent.
- (a) State and Prove Rolle’s Theorem
(b) Find a suitable c of Rolle’s Theorem for the function
.
- 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 |
- Find k
- Evaluate (a) (b) (c)
- If , find the minimum value of k
- Determine the distribution function of X.
- (a) If
is the joint p.d.f. of X and Y, find the marginal p.d.f.’s. Also, evaluate
P[ (X < 1) (Y < 3) ]
(b) Find the rank of the matrix .
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