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 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
PART – A
Answer ALL Questions: (10 x 2 =20)
- 1. Find the value of for which the function is a probability density function.
- Define: Correlation coefficient.
- Write down the density function of hyper geometric distribution.
- Obtain the mean of geometric distribution.
- Write down the density of bivariate normal distribution.
- Define: Chi-square statistic
- State any two properties of t-distribution
- Write down the distribution function of if .
- Obtain the density function of the nth(largest) order statistic when a sample of size n is drawn from a
population with pdf
- Define: Stochastic convergence.
PART – B
Answer any FIVE Questions: (5 x 8 =40)
- Let and have the joint pdf described as follows:
Obtain the marginal probability density functions and the conditional expectations
- Let and have the joint pdf
Examine whether the random variables are independent.
- Establish the lack of memory property of geometric distribution.
- State and prove the additive property of Binomial distribution.
- Find the median of Cauchy distribution with location parameter and scale paramter .
- Obtain the moment generating function of standard normal distribution.
- Show that ratio of two independent standard normal variates has Cauchy distribution.
- State and prove Central limit theorem for iid random variables.
PART – C
Answer any TWO Questions: (2 x 20 =40)
- (a) Derive the mean and variance of Poisson distribution.
(b) Let and have a bivariate binomial distribution with and
- (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.
(b) Let . Find the density function of
- (a) Derive the distribution of t-statistic.
(b) Derive the sampling distribution of sample mean from a normal population.
- (a) Find where is the largest order statistic based on a sample of size four from a
population with pdf
(b) Obtain the limiting distribution of nth order statistic based on a sample of size n drawn from
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