LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – NOVEMBER 2010

ST 3503/ST 3501/ST 3500 – STATISTICAL MATHEMATICS – II

 

 

 

Date : 30-10-10                     Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

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

 

1. Define upper and lower sums of a function on [a, b] corresponding to partition.
2. State the linearity property of integrals.
3. Define Gamma integral.
4. Evaluate
5. State the -test for an improper integral I kind.
6. State how the mean and variance are found from the m.g.f.
7. State the relationship between the characteristic roots and trace and determinant of a matrix.
8. Solve: .
9. Verify whether the following system of equations is consistent:

x – y + z – 2 = 0, 3x – y + 2z = 0,  3x + y + z +18 = 0.

1. Find the characteristics roots of .

 

PART – B

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

1. Evaluate  from first principles.
2. Show that if  is a monotonically increasing  function on [ a, b], then .
3. Discuss the convergence of for various value of p.
4. Solve .
5. Find the m.g.f and hence the mean and variance of a distribution with p.d.f.
6. Discuss the convergence of : (a)   (b)  (c) .
7. Evaluate the integralover the upper half of the circle.
8. Find any non trivial solution which may exist:

.

(P.T.O.)

 

 

 

 

 

PART – C

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

1. (a) If and , show that and that .

(b) State and prove the First Fundamental Theorem of Integral Calculus. Deduce the Second

Fundamental theorem.                                                                                                       (10 + 10)

 

1. (a) Show that .                                                                                                     (10)

(b) Show that mean does not exist for the distribution with p.d.f. .   (10)

1. a) If have jont pdf , find the p.d.f. of U=X₁/X₂.

(15)

1. b) Discuss the convergence of the integral for various values of a. (5)

1.  (a) State and prove Cayley-Hamilton Theorem.                                                                              (10)

 

(b) Find the inverse of the following matrix by using Cayley-Hamilton theorem.

.                                                                                                                       (10)

 

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