LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

YB 19

B.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – April 2009

ST 4501 – DISTRIBUTION THEORY

 

 

 

Date & Time: 24/04/2009 / 9:00 – 12:00        Dept. No.                                                       Max. : 100 Marks

 

 

PART – A

Answer ALL Questions                                                                                                   (10 x 2 =20)

1. Show that the function  is a probability density function.
2. Define : Conditional Variance with reference to a bivariate distribution.
3. Write the density function of discrete uniform distribution and obtain its mean.
4. Find the mean of binomial distribution using its moment generating function.
5. Find the maximum value of normal distribution with mean and variance.
6. Write down the density function of bivariate normal distribution.
7. Define : t Statistic.
8. Find the density function of the random variable where is the distribution function of the continuous random variable .
9. State central limit theorem for iid random variables.
10. Obtain the density function of first order statistic in.

 

PART – B

 

Answer any FIVE Questions                                                                                         (5 x 8 =40)

 

1. Let be iid RVs with common PDF . Write . Show that and are pairwise independent but not independent.
2. Let be independent RVs with common density given by

Find the distribution of .

1. Derive the characteristic function of Poisson distribution. Using the same find the first three central moments of Poisson distribution
2. Establish the lack of memory property of geometric distribution
3. Obtain the marginal distribution of X if (X,Y) follows bivariate normal distribution
4. Obtain the moment generating function of chi-square distribution and hence establish its additive property
5. Show that if , then
6. Derive the formula for the density function of rth order statistic

 

PART –  C

 

Answer any TWO Questions                                                                              (2 x 20 =40)

 

1. Let (X,Y) be jointly distributed with density

 

Find

 

1. (a) Obtain the moment generating function of Normal distribution and hence find its

mean and variance

 

(b) Show that, if X and Y are independent poisson variates with parameters and

,  then the conditional distribution of X given X+Y is binomial.

 

1. Derive the density function of t distribution and obtain its mean.

 

1. Write descriptive notes on the following :

• Stochastic independence
• Multinomial distribution
• Transformation of variables
• Stochastic Convergence

 

 

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