LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

YB 02

FIRST SEMESTER – April 2009

ST 1503/ ST 1501 – PROBABILITY AND RANDOM VARIABLES

 

 

 

Date & Time: 22/04/2009 / 1:00 – 4:00         Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

Answer ALL the questions:                                                             (10 x 2 = 20)

1. Define sample space and elementary event.
2. Four cards are drawn at random from a pack of 5 cards. Find the probability that two are kings and two are queens.
3. State the axioms of probability function.
4. For any two events A and B, show that .
5. Define pairwise and mutual independence of events.
6. The odds that a book in mathematics will be favourably reviewed by 3 independent critics are 3 to 2, 4 to 3 and 2 to 3 respectively. What is the probability that all the three reviews will be favourable?
7. If A and B are two mutually exclusive events show that .
8. A consignment of 15 record players contains 4 defectives. The record players are selected at random, one by one and examined. Those examined are not replaced. What is the probability that the 9^th one examined is the last defective?
9. Define a random variable.
10. Define probability generating function.

PART – B

Answer any FIVE questions:                                                                       (5 x 8 = 40)

11. Five salesmen A, B, C, D and E of a company are considered for a three member trade delegation to represent the company in an international trade conference. Construct the sample space and find the probability that i) A is selected ii) either A or B is selected.
12. For any three non-mutually exclusive events A, B and C, evaluate .
13. Compare the chances of throwing 4 with one die, 8 with two dice and 12 with three dice.
14. State and prove Boole’s inequality.
15. A and B are two students whose chances of solving a problem in statistics correctly are and If the probability of them making a common error is and they obtain the same answer, find the probability that their answer is correct.
16. Let the random variable X have the distribution þþ where þ. For what value of þ is the variance of X, a maximum.
17. The life (in 000’s of kms) of a car tyre is a random variable having the p.d.f.

Find the probability that one of these tyres will last i) at most 10,000 kms ii) at least 30,000 kms.

18. If X and Y are two random variables such that , then show that .

 

 

 

 

PART – C

Answer any TWO questions:                                                                       (2 x 20 = 40)

19. (a) State and prove the multiplication theorem of probability for any ‘n’ events.

(b) In a company, 60% of the employees are graduates. Of these 10% are in sales. Out of the employees who are not graduates, 80% are in sales. What is the probability that

1. i) an employee selected at random is in sales
2. ii) an employee selected at random is neither in sales nor a graduate?
3. a) State and prove Baye’s Theorem.
4. b) Two computers A and B are to be marketed. A salesman who is assigned the job of finding customers for them has 60% and 40% chances respectively of succeeding in case of computers A and B. The two computers can be sold independently. What is the probability that computer A is sold given that he was able to sell atleast one computer?
5. a) State and prove chebychev’s inequality
6. b) A symmetric die is thrown 600 times. Find the lower bound for the probability of getting 80 to 120 sixes.
7. The time a person has to wait for a bus is a random variable with the following p.d.f.

Let the event A be defined as a person waiting between 0 and 2 minutes (inclusive) and B be the event of waiting 0 and 3 minutes (inclusive)

1. Draw the graph of p.d.f
2. Show that and .

 

 

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