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:

(0,0) | (0,1) | (1,0) | (1,1) | (2,0) | (2,1) | |

1/18 | 3/18 | 4/18 | 3/18 | 6/18 | 1/18 |

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

. Obtain

- (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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