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```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
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
PART – A
1. Find the value of
(10 x 2 =20)
ce
for which the function f ( x)  
0
2 x
, if x  0
otherwise
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 X 2 if X ~ U (0,1) .
9. Obtain the density function of the nth(largest) order statistic when a sample of size n is drawn from a
e x , if x  0
.
population with pdf f ( x)  
otherwise
0
10. Define: Stochastic convergence.
PART – B
(5 x 8 =40)
11. Let X1 and X 2 have the joint pdf f ( x1, x2 ) described as follows:
( x1, x2 )
(0,0) (0,1) (1,0) (1,1) (2,0) (2,1)
f ( x1, x2 )
1/18
3/18
4/18
3/18
6/18
1/18
Obtain the marginal probability density functions and the conditional expectations
 x1  x2 if 0  x1  1,0  x2  1
12. Let X1 and X 2 have the joint pdf f ( x1 , x2 )  
otherwise
0
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 k .
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
(2 x 20 =40)
19. (a) Derive the mean and variance of Poisson distribution.
2
2
(b) Let X and Y have a bivariate binomial distribution with 1  3,  2  1,  1  16,  2  25 and
  0.6 . Obtain P(3  X  3 | Y  4).
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.
2
X
(b) Let X ~ N (  ,  ) . Find the density function of Y  e .
21. (a) Derive the distribution of t-statistic.
(b) Derive the sampling distribution of sample mean from a normal population.
22. (a) Find P(Y4  3) where Y4 is the largest order statistic based on a sample of size four from a
x
population with pdf f ( x)  e , x  0.
(b) Obtain the limiting distribution of nth order statistic based on a sample of size n drawn from
U (0, ),   0.
\$\$\$\$\$\$\$
```
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