LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

NO 12

THIRD SEMESTER – APRIL 2008

ST 3501 / 3500 – STATISTICAL MATHEMATICS – II

 

 

 

Date : 26/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

SECTION – A

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

 

1. Define Upper and Lower Sums of a function corresponding to a given partition of a closed interval.
2. State the Second Mean Value theorem for integrals.
3. If f(x) = C x² (1 – x), for 0 < x < 1 is a probability density function (p.d.f.), find the value of C.
4. Define improper integral of second kind.
5. Show that the improper integral  converges (where a > 0)
6. Find L^–1
7. State the general solution for the linear differential equation  + Py = Q
8. State the postulates of a Poisson Process.

 

1. State the Fundamental Theorem on a necessary and sufficient condition for the consistency of a system of equations  A+ = .
2. If λ is a characteristic root of A, show that λ² is a characteristic root of A².

 

SECTION – B

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

 

1. Let P_n = {0, 1/n, 2/n, ….., (n – 1)/n, 1} be a partition of [0, 1]. For the function f(x) = x, 0 ≤x ≤ 1, find U[P_n, f] and L[P_n, f]. Comment on the integrability of the function.
2. Evaluate: (i)  (ii)
3. Show that the integral (where a > 0) converges for p > 1 and diverges for p ≤ 1.

 

1. Define Beta integrals of First kind and Second kind. Show that one can be obtained from the other by a suitable transformation.

(P.T.O)

1. Solve:  =

 

1. Solve: (D² + 4D + 6) y = 5 e^{– 2 x}

 

1. Establish the relationship between the characteristic roots and the trace and determinant of a matrix.

 

1. Give a parametric form of solution to the following system of equations:

4x₁ – x₂ + 6x₃ = 0

2x₁ + 7x₂ +12x₃ = 0

x₁ – 4x₂ – 3x₃ = 0

5x₁ – 5x₂ +3x₃ = 0

 

SECTION – C

 

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

 

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

(b) Evaluate (i)  (ii)                                                      (12+8)

 

1. (a) Show that mean does not exist for the distribution with p.d.f.

f(x) = , – ∞ < x < ∞

(b)L[f(t)] = F(s), show that L[t f(t)] = – F(s). Using this result find L[t² e^{– 3 t}]

(8+12)

 

1. Evaluate: (a) ∫ ∫ x² y² dx dy over the circle x² + y² ≤ 1.

(b) ∫ ∫ y dx dy over the region between the parabola y = x² and the line x + y = 2

(10+10)

 

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

(b) If P is a non-singular matrix and A is any square matrix, show that A and      P^–1AP have the same characteristic equation. Also, show that if ‘x’ is a characteristic vector of A, then P^–1 x is a characteristic vector of P^–1AP.      (12+8)

 

 

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