LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AB 19

FIRST SEMESTER – NOV 2006

ST 1810 – ADVANCED DISTRIBUTION THEORY

(Also equivalent to ST 1806/1803)

 

 

Date & Time : 31-10-2006/1.00-4.00    Dept. No.                                                       Max. : 100 Marks

 

 

SECTION – A

Answer all the questions                                                                                  (10 x 2 = 20)

1. Define truncated distribution and give an example.
2. Show that geometric distribution satisfies lack of memory property.
3. Define bivariate binomial distribution.
4. If (X₁,X ₂) is bivariate Poisson, find the marginal distributions.
5. If (X₁,X ₂) is bivariate normal, find the distribution of X₁ – X ₂ .
6. Define bivariate exponential distribution of Marshall – Olkin.
7. Find the mean of non-central chi-square distribution.
8. Explain compound distribution.
9. Let X _{1 ,}X_{2 ,}X ₃ be independent N(0,1) random variables. Examine whether

X₁² + X₂ ² + 2X₃ ² – X₁X₂ + 2X₂X₃  has a chi-square distribution.

10. Let X _{1 ,}X_{2 ,}X ₃,X₄ be independent N(0,1) random variables. Find the MGF of X₁X₂+ X₃X₄.

 

SECTION – B

Answer any five questions                                                                                (5 x 8 = 40)

11. For a power series distribution, state and establish a recurrence relation satisfied by the

cumulants.

12. For a lognormal distribution, show that mean > median > mode.
13. State and establish the additive property for bivariate binomial distribution.
14. Derive the conditional distributions associated with bivariate Poisson distribution.
15. If X = (X₁,X ₂)^/ is bivariate normal with mean vector m and dispersion matrix S , then show that

a^/ X  and b^/ X are independent if and only if  a^/ S b = 0.

16. If X = (X₁,X ₂)^/ is bivariate exponential, find the distribution of Min{ X₁,X ₂}.
17. State and establish the additive property for noncentral chi-square distribution.
18. If X has N_p(m , S) distribution, then show that ( X – m )^/ S ^-1(X – m ) is distributed as chi-square.

 

SECTION – C

Answer any two questions                                                                                (2 x 20 = 40)   

19 a) State and establish a characterization of exponential distribution.

1. b) Let X₁, X₂, …,X_n denote a random sample from IG(m, l). Show that

 

1. =  S X_i /n  follows IG distribution
2. ii)  lV = l  (S 1/X_i – 1/  ) follows chi-square distribution

and   iii)        and V are independent.

 

20 a) State and establish a relation between bivariate binomial and bivariate Poisson distributios.

1. b) Define bivariate beta distribution.Derive its probability density function.

21 a) State and establish a characterization of bivariate exponential distribution.

1. b) Define non-central F distribution and derive its mean and variance.

22 a) State and prove Cochran theorem.

1. b) Given a random sample from normal distribution, using the theory of quadratic forms, show

that the sample mean and the sample variance are independent.

 

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