LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AC 05

SECOND SEMESTER – APRIL 2007

ST 2501 – STATISTICAL MATHEMATICS – I

 

 

 

Date & Time: 20/04/2007 / 1:00 – 4:00 Dept. No.                                                Max. : 100 Marks

 

 

SECTION – A

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

 

1. Define a bounded function and give an example.

 

1. State the values of and .
2. Write down the distribution function of the number of heads in two tosses of a fair coin.
3. Investigate the nature (convergence / divergence / oscillatory) of the series

1 – 2 + 3 – 4 + 5 – ∙∙∙∙∙∙∙∙

1. State the Leibnitz test for convergence of alternating series.

 

1. Apply first principles to find f ‘(a) for the function f(x) = xⁿ.

 

1. Show the validity of Rolle’s Theorem for f(x) = , x [– 1, 1].

 

1. Define a vector space.

 

1. If M(t₁,t₂) is the joint m.g.f. of  (X,Y), express E(X) and E(Y) in terms of M(t₁,t₂).

 

1. Define an Idempotent Matrix.

SECTION – B

 

 

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

1. Show that Inf f + Inf g  ≤  Inf (f + g)  ≤  Sup (f + g) ≤ Sup f + Sup g.
2. Show by using first principles that  = 0.
3. Discuss the convergence of the following series (a) ,  (b)
4. A discrete r.v. X has p.m.f. p(x) = , x = 0, 1, 2, …… Obtain the m.g.f. and hence mean and variance of X.

 

1. Show that differentiability implies continuity. Demonstrate clearly with an example that continuity does not imply differentiability.

 

1. Obtain the coefficients in the Taylor’s series expansion of a function about ‘c’.

(P.T.O)

 

 

 

 

1. State any two properties of a bivariate distribution function. If F(x , y)  is the bivariate d.f. of (X, Y), show that

P( a < X ≤ b , c < Y ≤ d) = F( b, d) – F ( b , c) – F( a, d) + F( a, c)

 

1. Establish the ‘Reversal Laws’ for the transpose and inverse of product of two matrices.

 

SECTION – C

 

 

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

 

1. (a) Establish the uniqueness of limit of a function as x → a (where ‘a’ is any real number). Also, show that if the limit is finite, then ‘f’ is bounded in a deleted neighbourhood of ‘a’.

(b) Identify the type of the r.v. X whose distribution function is

F ( x) =

Also, find P( X ≥ 4 / 3 ) and P(X ≤ 1).                                                          (12 + 8)

 

1. (a) Investigate the extreme values of the function f(x) = ( x + 5)²(x³ – 10)

(b) State the Generalized Mean Value Theorem. Examine its validity for the functions f(x) = x², g(x) = x⁴ for x[1, 2].                                                   (12 + 8)

 

1. (a) For the following function, show that the double limit at the origin does not exist but the repeated limits exist:

f (x ¸y) =

(b) Show that the mixed partial derivatives of the following function at the origin

are unequal:

f (x , y) =              (8 + 12)

 

1. (a) Show that if two (non-zero) vectors are orthogonal to each other, they are

linearly independent.

(b) Find the inverse of the following matrix by ‘partitioning’ method:

(5 + 15)

 

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