LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2012

ST 4502/ST 4501 – DISTRIBUTION THEORY

 

 

 

Date : 21-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL Questions:                                                                                                       (10 x 2 =20)

1. 1. Find the value of for which the function is a probability density function.
2. Define: Correlation coefficient.
3. Write down the density function of hyper geometric distribution.
4. Obtain the mean of geometric distribution.
5. Write down the density of bivariate normal distribution.
6. Define: Chi-square statistic
7. State any two properties of t-distribution
8. Write down the distribution function of if .
9. Obtain the density function of the nth(largest) order statistic when a sample of size n is drawn from a

population with pdf

10. Define: Stochastic convergence.

PART – B

 

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

 

11. Let and have the joint pdf  described as follows:

	(0,0)	(0,1)	(1,0)	(1,1)	(2,0)	(2,1)
	1/18	3/18	4/18	3/18	6/18	1/18

 

Obtain the marginal probability density functions and the conditional expectations

12. Let and have the joint pdf

Examine whether the random variables are independent.

13. Establish the lack of memory property of geometric distribution.
14. State and prove the additive property of Binomial distribution.
15. Find the median of Cauchy distribution with location parameter and scale paramter  .
16. Obtain the moment generating function of standard normal distribution.
17. Show that ratio of two independent standard normal variates has Cauchy distribution.
18. State and prove Central limit theorem for iid random variables.

PART – C

 

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

 

19. (a) Derive the mean and variance of Poisson distribution.

(b) Let and have a bivariate binomial distribution with  and

. Obtain

 

20. (a) Write down the density function of two parameter gamma distribution. Derive its moment

generating function and hence the mean and variance of the distribution.

(b) Let  . Find the density function of

 

21. (a) Derive the distribution of t-statistic.

(b) Derive the sampling distribution of sample mean from a normal population.

 

22. (a) Find where  is the largest order statistic based on a sample of size four from a

population with  pdf

(b) Obtain the limiting distribution of nth order statistic based on a sample of size n drawn from

 

 

 

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