LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION –STATISTICS
SUPPLEMENTARY EXAMINATION – JUNE 2007
ST 4501 - DISTRIBUTION THEORY
Date & Time: 26/06/2007 / 9:00 - 12:00
Dept. No.
Max. : 100 Marks
SECTION A
Answer ALL the questions
1. Define distribution function of a random variable ‘X’.
(10x2=20)
2. Define marginal distributions of the random variables X and Y.
3. What is meant by ‘Mutual Independence’ for a set of ‘n’ random variables?
4. Define conditional mean of a random variable X given a Y=y.
5. Determine mode of the following probability distribution,
p(x) = (4cx) (1/4)x (3/4)4 – x , x = 0 , 1, 2, 3, 4.
6. State the Condition(s) under which Binomial distribution tends to Poisson distribution.
7. Define Negative Binomial Distribution with usual notation and give an example
8. Define Student’s ‘t’ distribution with ‘v’ degrees of freedom
9. Show that Cauchy distribution is a particular case of student’s ‘t’ distribution.
10. Define ‘Order Statistics’.
SECTION B
Answer any FIVE questions only
(5x8 =40)
 x1  x 2

, x1  1,2,3; x 2  1,2
11. The joint p.d.f. of X1 and X2 is f(x1, x2) =  21
.

0, otherwise
[a]Find the Marginal p.d.f of X1 and X2 [b] Find P[X1=3] and P[X2=2].
12. Show that E(X| Y=y) = (n-y) p1/ (1-p2) in Trinomial distribution.
13. Prove that Hyper-Geometric distribution tends to Binomial distribution when N-> and
M/N = p .
14. Show that median and mode of Normal distribution are .
15. Derive the recurrence relation for the probabilities of Binomial Distribution.
16. Derive M.G.F. of Laplace distribution and hence find its mean and variance.
17. Derive the p.d.f of F-variate with (n1, n2) d.f.
18. Obtain Mean and Variance of Beta distribution of IInd kind.
SECTION C
Answer any TWO questions only
(2x20 =40)
2 , 0  x  y  1
19.[a]The random variables X and Y have the joint p.d.f. f(x, y) = 
.
 0, otherwise
Show that the correlation co-efficient between X and Y is ½.
[15]
[b] Let X and Y be two random variables with the p.d.f.
 x  y , 0  x  1,0  y  1
f(x, y) = 
,
0, otherwise

Examine whether X and Y are stochastically dependent or not.
[5]
20 [a] If X and Y are two independent Chi-Square variates with parameters n1 and n2
respectively. find the distribution of U = X / (X+Y).
[12]
[b] Obtain rth order raw moment expression for Uniform distribution and hence obtain
Mean and Variance.
[8]
21 [a]Obtain marginal distribution of X and conditional distribution of Y|X=x for
Bi-variate Normal distribution and hence derive it’s M.G.F.
[b] Derive M.G.F. of Poisson distribution and hence find its Mean and Variance.
[15]
[5]
22. [a] . Let Y1, Y2, and Y3 denote the order statistics of a random sample of size 3 from a
 1,
distribution having p.d.f. f(x) = 
0,
0  xi  1, i  1,2,3
.
Otherwise
Find the p.d.f. of sample range Z= Y3 - Y1
[b] Derive the p.d.f of Student’s ‘t’-variate with (v) d.f.
[8]
[12]
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