LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

NO 5

SECOND SEMESTER – APRIL 2008

ST 2501 – STATISTICAL MATHEMATICS – I

 

 

 

Date : 23/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION – A

Answer ALL the Questions                                                                    (10 x 2 = 20 marks)

 

1. Define a bounded function and give an example.
2. Write the formula for a_n for the sequence 1, –3, 5, –7, . . . .
3. State any two properties of a distribution function.
4. Give reason (state the relevant result) as to why the series  is divergent.
5. State the Limit Form of Comparison Test.
6. For the function f(x) = | x |, x  R, find f ‘(0 +) and f ‘(0 –).
7. Define ‘Stationary Point’ of a function.
8. Is the space V   = { (x₁, x₂, 2x₁ – x₂) | x₁, x₂  R }a vector subspace of R³ ? Justify your answer.
9. Define symmetric matrix
10. Define rank of a matrix.

 

SECTION – B

Answer any FIVE Questions                                                                    (5 x 8 = 40 marks)

 

1. Show that the function f(x) = xⁿ is continuous at every point of R
2. Show that a convergent sequence is bounded. Give an example to show that the converse is not true.
3. Show that the series 1 + + + + · · · · · ·    is convergent
4. Find the m.g.f. of the random variable with p.m.f. p(x) = p q^x, x = 0, 1, 2, ….

Hence find the mean.

1. Verify the Mean Value theorem for the function f(x) = x² +3x – 4   on the interval [1, 3].
2. Examine the continuity of the following function at the origin (by using first principles):

f(x) =

1. Verify whether the vectors [2, –1, 1]’, [1, 2, –1]’, [1, 1,–2]’ are linearly independent or dependent.
2. Find the rank of the matrix

(P.T.O)

 

 

SECTION – C

Answer any TWO Questions                                                                 (2x 20 = 40 marks)

 

1. (a) If f(x) = ℓ₁ and g (x) = ℓ₂ ≠ 0, then show that  = ℓ₁ / ℓ₂

(b) If p(x) = x / 15,  x = 1, 2, 3, 4, 5 be the probability mass function of a random variable X, obtain the distribution function of X.                                          (12+8)

 

1. (a) State the Cauchy’s condensation Test and using it discuss the convergence of the series for variations in ‘p’.

(b) Check whether the following series are conditionally convergent / absolutely convergent / divergent:

(i)   (ii)                                           (10+10)

 

1. (a) Obtain the Maclaurin’s Series expansion for the function f(x) = log (1 + x). Show that the expansion indeed converges to the function for –1 < x < 1 by analyzing the behaviour of the remainder term(s).

(b) Discuss the extreme values of the function f(x) = 2 x³ – 15 x² + 36 x + 1

(14+6)

 

1. (a) Establish the uniqueness of the inverse of a non-singular matrix. Also, establish the ‘Reversal Law’ for the inverse of product of two matrices.

(b) Find the inverse of the following matrix by using any method:

(8+12)

 

 

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