LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 33

FIRST SEMESTER – April 2009

ST 1810 – ADVANCED DISTRIBUTION THEORY

 

 

 

Date & Time: 28/04/2009 / 1:00 – 4:00  Dept. No.                                                    Max. : 100 Marks

 

 

SECTION – A                       Answer all the questions                                     (10 x 2 = 20)

 

1. Find the mean of truncated binomial distribution, truncated at 0.
2. Show that Posson distribution is a power series distribution
3. Define lognormal distribution and show that the square of a lognormal variable is also lognormal.
4. Show that the geometric distribution satisfies lack of memory property.
5. Find the mean of X₁X₂ when (X₁, X₂) has a bivariate Poisson distribution.
6. Let (X₁, X₂) have a bivariate binomial distribution. Find the distribution of X₁+X₂.
7. Define bivariate lack of memory property..
8. State the MGF associated with the bivariate normal distribution. Hence find the marginal

distributions.

9. Let X_1, X₂, X₃, X₄ be independent standard normal variables. Examine whether

2X₁² + 5 X₂² + X₃² +4 X₄² – 2 X₁X₂ + 4 X₂X₃ + 4 X₁X₄ is distributed as chi-square.

10. Let X be B( 2,q), q = 0.2, 0.3. If q is discrete uniform, find the mean of the compound

distribution.

 

SECTION – B                                Answer any five questions                           (5 x 8 = 40)

 

11. State and establish a characterization of geometric distribution based on order statistics. 12. Find the

conditional distributions associated with trinomial distribution.

13. If (X_1, X₂) is Bivariate Poisson, show that marginal distributions are Poisson.
14. Derive the MGF of inverse Gaussian distribution. Hence find the mean and the variance.
15. State and establish the relation between the mean, the median and the mode of lognormal

distribution.

16. If (X_1, X₂) is Bivariate exponential, show that min{X₁,X₂}is exponential
17. Find the mean and variance of non-central chi-square distribution.
18. Given a random sample from a normal distribution, show that the sample mean and the sample

variance are independent, using the theory of quadratic forms.

 

SECTION – C                               Answer any two questions                         (2 x 20 = 40)

 

19 a) State and establish the  characterization of exponential distribution based on lack of memory

property.

1. b) If (X_1, X₂) is Bivariate normal, state and establish a necessary and sufficient condition for two

linear combinations of X₁ and X₂  to be independent.

20 a) State and establish the additive property of  bivariate Poisson distribution.

1. b) State and establish a characterization of Marshall-Olkin bivariate exponential distribution.

21 a) Define non-central t- variable and derive its pdf.

1. b) State and establish the additive property of non-central chi-square distribution.

22 a) Let X be distributed as multivariate normal with mean vector µ and the dispersion matrix Σ. Show that

(X – µ )^/ Σ ^-1(X – µ ) is distributed as chi-square.

1. b) State and establish Cochran’s theorem on quadratic forms.

 

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