Loyola College B.Sc. Statistics April 2008 Statistical Mathematics – II Question Paper PDF Download

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 Pn = {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[Pn, f] and L[Pn, 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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