B.Sc. DEGREE EXAMINATION – STATISTICS
SIXTH SEMESTER – SUPPLEMENTARY – JUNE 2012
ST 4502/ST 4501 - DISTRIBUTION THEORY
Date : 29-06-2012
Time : 2:00 - 5:00
Dept. No. Max. : 100 Marks
PART - A
Answer ALL Questions (10X2=20)
1.
Suppose that two dimensional continuous random variable ( X , Y ) has joint p.d.f given by
2.
 

6
0
2 x y , 0 x otherwise
Let X be a random variable with mean
1, 0
 y 1
. Find and variance
 f (x / y)
2 . Show that of b , is minimum when b =

3.
If X
1 and X
2
are i.i.d. Geometric variates, find the distribution of
E
X
1

X
2
(
.
X
 b )
2
, as function
4.
Obtain the mean of Beta distribution of I kind.
5.
State the important properties of Normal distribution.
6.
Obtain the mean of Laplace distribution.
7.
Find the mean of uniform distribution over the interval (a, b).
8.
Derive the distribution of n th order statistic.
9.
If F is a r.v. following F ( n
1
, n
2
) , find the distribution of 1 / F.
10.
Define Stochastic convergence.
PART –B
Answer any FIVE Questions (5X8=40)
11.
The joint probability distribution of two random variables X and Y is given by
P ( X

0 , Y

1 )

1 / 3 , P ( X

1 , Y
 
1 )

1 / 3 and P ( X

1 , Y

1 )

1 / 3
Find (a) marginal distribution of X and Y . (b) Conditional distribution of X given Y=1.
12.
Find the moment generating function of binomial distribution with parameters n and p and hence find its mean and variance.
13.
Prove that Exponential distribution has ‘Lack of Memory’ property.
[P. T. O]
14.
Let X and Y be independent Gamma variates. Find the joint p.d.f. of U = X + Y and V = X /
(X+Y). Hence, find the marginal p.d.f. of V.
15.
Obtain the marginal distributions of X and Y in the case of bivariate normal distribution.
16.
Let X and Y are two independent standard normal variables. Find distribution of X/Y and identify it.
17.
State and prove Central limit theorem for i.i.d. random variables.
18.
Find mean, standard deviation and mode of Chi-square distribution.

PART – C
Answer any TWO Questions. (2X20=40)
19.
(a) Derive the m.g.f. of Negative Binomial distribution. Hence, find its mean and variance.
(b) Find the moment generating function trinomial distribution and hence find the
correlation between the two variables.
20.
(a) Prove that for a Normal distribution all odd order central moments are zero and find the expression for even order moments.
(b) Find the mode of Poisson distribution.
21.
(a) Define the Hyper-geometric distribution. Find its mean and variance.
(b) Derive the p.d.f. of t distribution.
22.
Identify the distribution of sample mean and sample variance. Also prove that they are independently distributed. Assume the parent population is Normal.
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