LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
FOURTH SEMESTER – April 2009
YB 19
ST 4501 - DISTRIBUTION THEORY
Date & Time: 24/04/2009 / 9:00 - 12:00
Dept. No.
Max. : 100 Marks
PART - A
Answer ALL Questions
1.
2.
3.
4.
(10 x 2 =20)
1 | x |
Show that the function f ( x)  e ,    x   is a probability density function.
2
Define : Conditional Variance with reference to a bivariate distribution.
Write the density function of discrete uniform distribution and obtain its mean.
Find the mean of binomial distribution using its moment generating function.
Find the maximum value of normal distribution with mean  and variance  2 .
Write down the density function of bivariate normal distribution.
Define : t Statistic.
Find the density function of the random variable Y  F ( X ) where F is the distribution function
of the continuous random variable X .
9. State central limit theorem for iid random variables.
10. Obtain the density function of first order statistic in U (0, ) .
5.
6.
7.
8.
PART - B
Answer any FIVE Questions
(5 x 8 =40)
1
. Write X 3  X 1 X 2 . Show that
2
X 1 , X 2 and X 3 are pairwise independent but not independent.
11. Let X 1 , X 2 be iid RVs with common PDF P[ X  1] 
1 if 0  x  1
12. Let X 1 , X 2 be independent RVs with common density given by f ( x)  
0 otherwise
Find the distribution of Y  X1  X 2 .
13. Derive the characteristic function of Poisson distribution. Using the same find the first three
central moments of Poisson distribution
14. Establish the lack of memory property of geometric distribution
15. Obtain the marginal distribution of X if (X,Y) follows bivariate normal distribution
16. Obtain the moment generating function of chi-square distribution and hence establish its additive
property
1
~ F ( n, m )
17. Show that if X ~ F (m, n) , then
X
18. Derive the formula for the density function of rth order statistic
PART - C
Answer any TWO Questions
(2 x 20 =40)
19. Let (X,Y) be jointly distributed with density
1

 y(1  x)  4 e  y (1 x) if x, y  0
f ( x, y)  

0 otherwise
Find E ( X ), E ( X | Y ), V ( X ), V ( X | Y )
20. (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 1 and
2 , then the conditional distribution of X given X+Y is binomial.
21. Derive the density function of t distribution and obtain its mean.
22. Write descriptive notes on the following :
(a) Stochastic independence
(b) Multinomial distribution
(c) Transformation of variables
(d) Stochastic Convergence
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