LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

YB 15

FOURTH SEMESTER – April 2009

ST 4206/ ST 4201 – MATHEMATICAL STATISTICS

 

 

 

Date & Time: 27/04/2009 / 1:00 – 4:00  Dept. No.                                                        Max. : 100 Marks

 

 

                                                                        PART – A 

                                                     

ANSWER ALL THE QUESTIONS                                                                          (10 x 2 = 20)

 

1. Define Mutually Exclusive Events with examples.
2. State multiplication law of probability
3. What is the chance that a leap year selected at random will contain 53 sundays?
4. Define probability generating function of a random variable.
5. If X is a random variable and a and b are constants, then show that

E (aX + b) = a E(X) + b provided all the expectations exist.

6. Derive the moment generating function of Poisson distribution.
7. Define Beta Distribution of First kind.
8. Define Regression.
9. Ten unbiased coins are tossed simultaneously. Find the probability of getting at least seven heads.
10. Define Most Powerful test.

 

PART – B

ANSWER ANY FIVE QUESTIONS                                                                          (5 x 8 = 40)

11. Four cards are drawn at random from a pack of 52 cards. Find the probability that
12. These are a king, a queen, a jack and an ace.
13. Two are kings and two are queens
14. Two are black and two are red.
15. There are two cards of hearts and two cards of diamonds.

 

12. The contents of urns I, II and III are as follows:

Urn I   : 1 White, 2 Black and 3 Red balls

Urn II  : 2 White, 1 Black and 1 Red balls, and

Urn III : 4 White, 5 Black and 3 Red balls

One urn is chosen at random and two balls drawn from it. They happen to be white and              red. What is the probability that they come from Urns I, II or III?

13. Let X be a Continuous random variable with probability density function

 

a). Determine the constant a

b). Compute P( X ≤ 1.5 )

 

14. State and prove Chebyshev’s Inequality.

 

15. Derive the Mean and Variance of Binomial Distribution.

 

16. The joint probability distribution of two random variables X and Y is given by

P ( X = 0, Y = 1) =  , P ( X = 1, Y = -1) = and P ( X = 1, Y = 1) =. Find

1. Marginal distributions of X and Y and
2. Conditional probability distribution of X given Y = 1.

 

17. The following figures show the distribution of digits in numbers chosen at random

from a telephone directory:

Digits	0	1	2	3	4	5	6	7	8	9	Total
Frequency	1026	1107	997	966	1075	933	1107	972	964	853	10,000

 

Test whether the digits may be taken to occur with equal frequency in the directory.

18. Define
    1. Null Hypothesis
    2. Alternative Hypothesis

• Level of Significance

1. Two types of Errors                                                     ( 2 + 2 + 2 + 2 )

 

PART – C

ANSWER ANY TWO QUESTIONS                                                                        (20 x 2 = 40)

19. (a). State and Prove Baye’s Theorem . (10)

 

(b). Three groups of Children contain respectively 3 girls and 1 boy, 2 girls and 2 boys,

and 1 girl and 3 boys. One child is selected at random from each group. Show that

the chance that the three selected consist of 1 girl and 2 boys is  .               (10)

 

20. (a). In four tosses of a coin, let X be the number of heads. Tabulate the 16 possible

outcomes with the corresponding values of X. By simple counting, derive the

probability distribution of X and hence calculate the expected value of X        (10)

 

 

 

(b). A random variable X has the following probability density function:

 

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

 

1. Find k
2. Evaluate

• If , find the minimum value of a.

1. Determine the distribution function of X (10)

 

21. (a). State any five properties of Normal Distribution (8)

 

(b). A manufacturer, who produces medicine bottles, finds that 0.1% of the bottles

are defective. The bottles are packed in boxes containing 500 bottles. A drug

manufacturer buys 100 boxes from the producer of bottles. Using Poisson

distribution, find how many boxes will contain :

1. no defective and
2. at least two defectives                                                                             (12)

 

22. (a). If , find
    1. Var (X)
    2. Var (Y)

• r (X,Y) (10)

(b). The mean weekly sales of soap bar in departmental stores was 146.3 bars per store.

After an advertising campaign the mean weekly sales in 22 stores for a typical week

increased to 153.7 and showed a standards deviation of 17.2. Was the advertising

campaign successful?                                                                                           (10)

 

 

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