LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

AB 02

FIRST SEMESTER – NOV 2006

ST 1501 – PROBABILITY AND RANDOM VARIABLES

 

 

Date & Time : 03-11-2006/1.00-4.00     Dept. No.                                                       Max. : 100 Marks

 

 

SECTION – A

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

 

1. Define Mutually Exclusive Events and give an example.
2. Using the Axioms of Probability, prove that if AB, P(A) ≤ P(B).
3. Draw Venn Diagram to represent the occurrence of at least two of three events A, B, C.
4. State the number of ways in which a set of 10 objects can be partitioned into three subsets of sizes 5, 3 and 2 respectively.
5. Explain the ‘Matching Problem’ and find the probability that a specified match occurs.
6. In an experiment of forming three-letter English words with three distinct letters, what is the probability that the first and third letters are consonants while the second one is a vowel?
7. Four statisticians arrange to meet at Hotel Five Star in a city. But there are four hotels in the city with the same name. If each one randomly chooses a hotel, what is the probability that they will all choose different hotels?
8. In tossing a coin thrice, what is the conditional probability that the first toss results in Head given that there are two Heads?
9. If A and B are independent events, show that A and B^c are independent.
10. Define a Continuous random variable.

SECTION – B

 

 

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

 

1. State the Binomial Theorem. Using Pascal’s Triangle write down the expansion of (a + b)⁶.

 

1. A study of 1000 people revealed that 500 were successful in their careers, 300 had studied Statistics and 200 had both studied Statistics and were successful in their careers. Find the probability that a randomly chosen person from the lot

• neither studied Statistics nor is successful in his career.
• is successful in his career but had not studied Statistics.
• had studied Statistics but is not successful in his career.

 

1. Two fair dice are thrown. Find the probability that (a) at least one of the dice shows up an odd number; (b) the sum of the two numbers exceeds 7.           (4 + 4)

 

1. In a random rearrangement of the letters of the word COMMERCE, find the probability that

• All the vowels come together
• All the vowels occupy odd number positions. (4 + 4 )

 

1. Show that P(A | B ) + P (A^c | B) = 1.  Give an example to show that in general  P(A | B) + P(A | B^c) ≠ 1.

 

1. An urn contains 12 balls out of which 8 are white. A sample of size 4 is drawn one by one. Find the conditional probability that the first ball drawn is white given that the sample contained two white balls. Solve this under with replacement and without replacement selections.

 

 

 

 

 

1. A number is drawn at random from 1 to 4. Discuss the independence of the following three events:

A: Number is < 3,  B: Number is odd,  C: Number is 1 or 4

 

1. State and prove the ‘Multiplication Theorem of Probability’ for many events.

 

SECTION – C

 

 

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

 

1. (a) State the Addition Theorem of Probability for two events. Hence state and prove the same for three events.

(b) Let A and B be events. For k = 0, 1, 2, express the following probabilities in

terms of P(A), P(B) and P(AB):

• P( exactly ‘k’ of the events A and B occur)
• P(at least ‘k’ of the events A and B occur) (8 +12)

 

1. An urn contains 5 white and 4 black balls from which 3 balls are drawn. Find the probability that (i) all three are black, (ii) at least two are white. Solve this under with replacement, without replacement and subset (unordered) selection schemes.

 

1. (a) State and prove the ‘Law of Total Probability’. Hence establish Baye’s Theorem.

(b) Three persons A₁, A₂, A₃ compete for the position of finance Managers of a company. The probability for A₁ to get appointed is 3/8, for A₂ it is 1/2 and for A₃ it is 1/8. If A₁ becomes the manager, the probability that he will introduce a Bonus scheme is 3/10. The corresponding probabilities in the case of A₂ and A₃ are 1/3 and 3/5. Given that the Bonus scheme has been introduced, what is the probability that A₁ was appointed?                                                                                 (10 +10)

 

1. (a) A random variable X has the following p.m.f.:

x	-2	-1	0	1	2
P(X =x)	1/10	1/5	2/5	1/5	1/10

 

 

 

Find the c.d.f. of X. Also. Find the mean and variance of X.

(b) If f(x) = cx², – 1< x < 2 , is the p.d.f. of a continuous random variable X, find

the value of ‘c’ and evaluate P( 0 < X < 1).                                                (13 +7)

 

 

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