LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

B.Sc., DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – NOVEMBER 2003

ST-2500/STA501 – STATISTICAL MATHEMATICS – I

01.11.2003                                                                                                           Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL the questions.                                                                              (10×2=20 marks)

 

1. Define ‘permutation with indistinguishable objects’.  State its value in factorial notation.
2. If A, B, C are events, write the set notation for the following: (i) A or B but not C occur  (ii) None of the three events occur.
3. If A and B are independent events, show that A and B^C are independent.
4. What are the supremum and infinum of the function f(x) = x – [x], x ÎR
5. Define probability mass function (p.m.f) of a discrete random variable and state its properties.
6. “The series is divergent”  – justify.
7. ” The function f (s) = is not a probability generating function” (p.g.f)” – justify.
8. For what value of ‘a’ does the sequence {4aⁿ}define a probability distribution on the set of positive integers?
9. Find f ^‘(3) for the function f(x) = , xÎ
10. Define radius of convergence of a power series.

 

SECTION-B

Answer any FIVE the questions.                                                                     (5×8=40 marks)

 

11. In how many ways 3 Americans, 4 French, 4 Germans and 2 Indians be seated in a row so that those of the same nationality are seated together? Find the number  of ways, if they are seated around a circular table.
12. State and prove the Addition Theorem of Probability for two events. Extend it for three events.
13. a) Show that limit of a convergent sequence is unique.
14. b) Define monotonic sequence with an example.
15. Consider the experiment of tossing a fair coin indefinitely until a head appears. Let X = Number of tosses until first Head.  Write down the p.m.f. and c.d.f of X.
16. Discuss the convergence of the Geometric series for variations in ‘a’.
17. Investigate the extreme values of the function f(x) = 2x³-3x²-36x+10, xÎ
18. Show that the series is divergent.
19. Define Binomial distribution and obtain its moment Generating function (m.g.f). Hence find its mean and variance.

 

 

 

SECTION-C

Answer any TWO the questions.                                                                     (2×20=40 marks)

 

19. a) Establish the theorem on Total probability.
20. b) Establish Baye’s theorem
21. c) Three machines produce 50%, 30% and 20% of the total products of a factory. The percentage of defectives manufactured by these machines are 3%, 4% and 5% of their total output. If an item is selected at random from the items produced in the factory, find the probability that the item is defective.  Also given that a selected item is defective. What is the probability that it was produced by the third machine?                       (6+6+8)
22. a) Show by using first principle that the function f(x) = x² is continuous at all points of R.
23. b) Identify the type of the r.v. whose c.d.f. is

0,        x < 0

f(x) =      x/3,    0 £ x < 1

2/3,    1 £ x < 2

1,       2 £  x

 

Also find P (X = 1.5), P(x < 1),  P (1£ x £ 2), P (x ³ 2) P (1 £ x < 3).                     (8+12)

21. a) Test the convergence of (i) (ii)

For each case, state the ‘test’ which you use.

1. b) Identify the probability distribution for which f (s) =is the p.g.f.  Find

the Mean and Variance of the distribution.                                                           (10+10)

22. a) Verify the applicability and validity of Mean Value theorem for the function

f(x) = x (x-1) (x-2),  xÎ[0,1/2].

1. b) Obtain the expansion of the Exponential function and hence define Poisson

distribution.                                                                                                            (10+10)

 

 

Go To Main page

 

 

 

Latest Govt Job & Exam Updates:

View Full List ...