LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

BA 07

THIRD SEMESTER – November 2008

ST 3500 – 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 corresponding to a partition of an interval [a, b].
2. State the linearity property of Riemann integrals.
3. Define probability density function (p.d.f.).
4. What is an improper integral? Give an example.
5. Define absolute convergence of an improper integral.
6. Is the integral  dx convergent or divergent?
7. Change the order of integration in the double integral
8. Define a symmetric matrix.
9. Show that inverse of a non-singular matrix is unique.
10. Define characteristic root and vector of a matrix.

 

SECTION – B

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

 

1. If P  is any partition of [a, b], show under usual notations that

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

 

1. Show that if f is R-integrable on [a, b], then | f | is integrable on [a, b].

 

1. If f(x) = cx , 0 < x < 1, is a p.d.f. find ‘c’ and the mean and variance of the distribution.
2. Discuss the convergence of the improper integral  dx (where a > 0) by varying ‘p’.

 

1. Evaluate ∫ ∫ xy dy dx over the positive quadrant of the circle x² + y² = a²

 

1. Define moment generating function of a bivariate distribution. Show how the means, variances and covariance can be found from it.

 

1. If λ is the characteristic root of a matrix A, show that λⁿ is a characteristic root of Aⁿ and both have the same associated characteristic vector. Also, show that one can always find a normalized (unit) characteristic vector associated with a characteristic root.

 

1. Define (i) Hermitian matrix, (ii) Idempotent matrix, (iii) Scalar matrix,    (iv) Orthogonal matrix

 

 

 

 

 

SECTION – C

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

 

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

 

1. (a) State and prove the Comparison Test for convergence of an improper integral of any one kind.

(b) Test the convergence of the integrals: (i)  dx  (ii)  dx          

(10 +10)

 

1. Let f(x, y) = x + y, 0 < x, y < 1, be the joint p.d.f. of (X, Y). Find the joint distribution function. Also, find the means, variances and covariance.

 

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

(b) Find the inverse of the following matrix using the above theorem:

(10 +1 0)

 

 

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