LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AK 37

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2008

    ST 1810 – ADVANCED DISTRIBUTION THEORY

 

 

 

Date : 03/05/2008            Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

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

 

1. Define zero truncated binomial distribution and find its mean.
2. Show that geometric distribution obeys lack of memory property.
3. If X is lognormal, show that X² is lognormal.
4. Define bivariate Poisson distribution.
5. Let (X₁, X₂) have a bivariate binomial distribution. Show that the marginals are binomial.
6. State bivariate lack of memory property.
7. Let (X₁, X₂) have a bivariate normal distribution. Show that X₁ – X₂ has a normal distribution.
8. Define non-central F distribution.
9. Let X₁, X₂, X₃, X₄ be independent standard normal variables. Examine whether

(X₁² + 2X₂² + X₃² + 2X₄² – 2 X₁X₂ + 2X₂X₃ )/ 2 is distributed as chi-square.

10. Let X have B(1,p), where p is uniform on (0,1). Show that the compound distribution is uniform.

 

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

 

11. Express the following distribution function as a mixture and hence find its mgf.

F(x)  =  0,                 x < 0

=  (x + 2)/8,     0 ≤ x  < 1

=   1,                x  ≥  1.

12. State and establish the additive property satisfied by bivariate binomial distribution.
13. Show that the regression equations are linear in the case of bivariate Poisson distribution.
14. For lognormal distribution, show that mean > median > mode.
15. State the mgf of Inverse Gaussian distribution. Hence find the cumulants.
16. State and establish any two properties of Pareto distribution.
17. Let X₁, X₂, X₃ be independent standard normal variables. Find the mgf of

X₁² + 5X₂² + 4X₃²  – 2 X₁X₂ + 8X₂X₃ .

18. Let X be N(,). Show that (X – )^{/ -1} (X – ) has a chi-square distribution.

 

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

 

19 a) Let X₁, X₂ be independent and identically distributed non-negative integer-valued random variables.

Show that X₁ given X₁ + X₂ is uniform if and only if X₁ is geometric.

1. b) Derive the pgf of the power series class of distributions. Hence find the pgf of binomial and Poisson

distributions.

20 a) Let X₁ and X₂  be independent and identically distributed random variables with finite second

moment.

Show that X₁ + X₂  and X₁ – X₂ are independent if and only if X₁ is normal.

1. b) Let X₁,X₂,…,X_n be independent random variables such that X_i has Inverse Gaussian distribution with

parameters _i, µ_i , I = 1,2,…,n. Show that

_i  (X_i – µ_i)² / (X_i µ_i²)  is chi-square with n degrees of freedom.

21 a) Define bivariate exponential distribution. Show that the marginals are exponential.

1. b) Derive the pdf of non-central chi-square distribution.

22 a) State and establish a necessary and sufficient condition for a quadratic form in N(0,1) variables to

have  a chi-square distribution.

1. b) Using the theory of quadratic forms, show that the sample mean and the sample variance are

independent in the case of normal distribution.

 

 

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