LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

BA 08

THIRD SEMESTER – November 2008

ST 3501 – STATISTICAL MATHEMATICS – II

 

 

 

Date : 06-11-08                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

 

 

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

 

1. Define lower sum of a function for a given partition of a closed interval [a, b].
2. Give an example of a non-monotonic function that is integrable.
3. Define probability density function (p.d.f.) of a random variable (r.v).
4. Give one example each for improper integrals of I and II kinds.
5. Show that the improper integral  converges absolutely.
6. Find the value of
7. If L[f(t)] = F(s), express L[ tⁿ F(t)] in terms of F.
8. State the general solution for the linear differential equation  + Py = Q
9. Define linearly dependent vectors.
10. Define a positive definite quadratic form.

 

SECTION – B

 

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

 

1. For any partition P of [a, b], show (under usual notations) that

m (b – a ) ≤ L(P , f)  ≤ U(P , f) ≤ M (b – a )

 

1. State (without proof) a necessary and sufficient condition for R-integrability of a function in a closed interval [a, b]. Using this condition, show that if g R[a, b] and if g is bounded away from 0, then 1/g  R[a, b]
2. If a r.v. X has p.d.f. f(x) = ,  – ∞ < x < ∞, find the moment generating function of X. Hence find its mean and variance.
3. Show that: (a)  is divergent (b)   is convergent
4. Find L[ cos at] and hence find L[ sin at] using the result for Laplace transform of the derivative of a function.

 

1. Solve: (x + y)² d x = 2 x² d y.

 

1. Find any non-trivial solution which exists for the following system of equations:

2 x₁ +3 x₂ – x₃ + x₄ = 0

3 x₁ + 2 x₂ – 2 x₃ + 2 x₄ = 0

5 x₁ – 4 x₃ + 4 x₄ = 0

 

1. Show that if λ is a characteristic root of A, then λⁿ is a characteristic vector of Aⁿ with the same associated characteristic vector. Establish a similar result for A^{– 1}.

 

SECTION – C

 

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

 

1. (a) State and prove the First and Second Fundamental Theorems of Integral Calculus.

 

(b) If f(x) = c x², 0 < x < 2, is the p.d.f. of a r.v X, find ‘c’ and P( 1 < X < 2)

(16+4)

1. (a) Show that  is divergent

(b) Discuss the convergence of gamma integral.                                                         (8+12)

 

1. (a) Evaluateover the region between the parabola y = x² and the line x+y =2.

(b) Evaluate  over the half circle x² + y² ≤ a² with y ≥ 0.

(10+10)

 

1. (a) State and prove Cayley-Hamilton Theorem.

 

(b) Show that a polynomial of degree ‘n’ has (n +1) distinct real roots if and  only

if all its coefficients are zero.                                                                                        (12+8)

 

 

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