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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
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 P(A  B)  P( B)  P( A  B) .
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?
 B  P( A){1 P(B) .
7. If A and B are two mutually exclusive events show that P A
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 9th 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 P( A  B  C ) .
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
1
1
1
and respectively. If the probability of them making a common error is
and
6
8
525
they obtain the same answer, find the probability that their answer is correct.
16. Let the random variable X have the distribution
P[ X  O]  P[ X  2] 
þ P[ X  1]  1  2 þ
where 0 
þ  12 . 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.
 1 e x 20

f ( x)   20
 0

YB 02

if x  0 

if x  0 
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 Y  X , then show that E (Y )  E ( X ) .
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
i) an employee selected at random is in sales
ii) an employee selected at random is neither in sales nor a graduate?
20. a) State and prove Baye’s Theorem.
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?
21. a) State and prove chebychev’s inequality
b) A symmetric die is thrown 600 times. Find the lower bound for the probability of
getting 80 to 120 sixes.
22. The time a person has to wait for a bus is a random variable with the following p.d.f.
for x  0
 0
1
 ( x  1)
9
4 
1
 x 
2
9 
 4 5
f ( x)     x 

9  2
1
 (4  x)
9
1

9
 0
for 0  x  1
for 1  x 
for
3
2
3
x2
2
for 2  x  3
for 3  x  6
for x  6
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)
i)
Draw the graph of p.d.f
ii)
 A  23 and P  A  B   13 .
Show that P B