LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

NO 2

           B.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2008

ST 1501 (PROBABILITY AND RANDOM VARIABLES)

 

 

 

Date : 06-05-08                  Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION –A

 

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

 

1. State the mathematical definition of probability.
2. A, B and C are three mutually exclusive and exhaustive events associated with a random experiment. Find given .
3. If A, B and C are three arbitrary events, find expressions for the events given below:

1. Both A and B but not C occur
2. At least one occur

1.  Two dice are tossed. Find the probability of getting an even number on the first die.
2. For any event , show that A and null even are independent.
3. The odds in favour of manager X settling the wage dispute with the workers are 6:8 while the odds in favour of manager Y settling the same dispute are 14:16. What is the probability that the dispute is settled?
4. Define ‘aprior’ and ‘posterior’ probabilities.
5. A university has to select an examiner from a list of 50 persons – 20 of them being women and 30 men, 10 of them know Tamil and 40 not, 15 of them are teachers while the remaining 35 are not. What is the probability that the University selects a Tamil knowing woman teacher?
6. Define a random variable.
7. If X is a random variable and ‘a’, ‘b’ are constants, then show that .

 

SECTION –B

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

 

1.  If , then show that

 

1. Twenty five books are placed at random in a shelf. Find the probability that a particular pair of books shall be (i) always together and (ii) never together.
2. If A₁, A₂, …, A_n are ‘n’ events, then show that
3. Three newspapers A, B and C are published in a certain city. It is estimated from a survey that, out of the adult population 20% read A, 16% read B, 14% read C, 8% read both A and B, 5% read both A and C, 4% read both B and C, 2% read all the three. Find what percentage read atleast one of the papers.
4. An urn contains four tickets marked with numbers 112, 121, 211, 222 and one ticket is drawn at random. Let A_i , i=1,2,3 be the event that the i^th digit of the number of the ticket drawn is 1. Discuss the independence of the events A₁, A₂ and A₃.

 

 

1. A random variable x has the following probability mass function:

x:	0	1	2	3	4	5	6	7
p(x):	0	k	2k	2k	3k	k2	2k2	7k2+k

1. i) Find k ii) Evaluate

1. Let X be a random variable with the following probability distribution:

x:	-3	6	9
p(x)	1/6	1/2	1/3

Find and hence .

1. State and prove multiplication theorem of probability for ‘n’ events.

SECTION –C

 

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

 

1. a) State and prove addition theorem of probability for 3 events.       (10)

1. b) Three groups of children contain 3 girls and 1 boy; 2 girls and 2 boys; 1 girl and 3 boys respectively. One child is selected at random from each group. Find the chance that the three selected consist of 1 girl and 2 boys. (10)

1. a) State and prove Baye’s theorem for future events.

1. b) The probability of x, y and z becoming managers are The probability that the Bonus scheme is introduced if x, y and z become mangers are , and respectively. i) What is the probability that the Bonus scheme will be introduced? ii) If the Bonus scheme has been introduced, what is the probability that the manager appointed was x?

1. Two dice, one green and the other red are thrown. Let A be the event that the sum of the points on the faces is odd and B be the event that atleast one number is 1.

1. Define the complete sample space and the events A, B, and
2. Find the probability of the events:

, ,,

1. a) State and prove chebychev’s inequality

1. b) Let the random variable x have the distribution:

.

For what value of p is the var (x) a maximum?

 

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