LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 25

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – APRIL 2007

ST 1810 / 1803 – ADVANCED DISTRIBUTION THEORY

 

 

 

Date & Time: 30/04/2007 / 1:00 – 4:00      Dept. No.                                       Max. : 100 Marks

 

 

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

1. Define truncated Poisson distribution and find its mean.
2. Show that the minimum of two independent exponential random variables is exponential.
3. Define bivariate Poisson distribution.
4. If (X₁,X ₂) is bivariate binomial, find the pgf of X₁+X ₂.
5. If (X₁,X ₂) is bivariate normal, find the distribution of 2X₁ -3 X ₂ .
6. Find the marginal distributions associated with a bivariate exponential distribution of

Marshall – Olkin.

7. Find the mean of non-central t – distribution.
8. Let X _{(1) ,}X_{(2 ),}X ₍₃₎ be order statistics from exponential distribution.Find E{X ₍₃₎ – X₍₁₎}.
9. Let X _{1 ,}X_{2 ,}X ₃ be independent N(0,1) random variables. Examine whether

2X₁² + X₂ ² + 2X₃ ² – X₁X₃ + 2X₂X₃  has a chi-square distribution.

10. Let X _{1 ,}X₂ be independent N(0,1) random variables. Find the MGF of 2X₁X₂.

 

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

11. State and establish the mgf of a Power series distribution.Deduce the mgf of Binomial.
12. For an Inverse Gaussian distribution,derive the cumulants. Hence find the mean and variance.
13. State and establish the additive property for bivariate Poisson distribution.
14. Derive the conditional distributions associated with bivariate binomial distribution.
15. Let X = (X₁,X₂)^/ be such that every linear combination of X₁and X₂ is distributed as normal.

Show that X is bivariate normal.

16. If X = (X₁,X ₂)^/ is bivariate exponential(l₁, l₂, l₃), show that X₁ and X ₂ are independent if and only

if  l₃ = 0.

17. Define non – central chi-square distribution and derive its probability density function.
18. If X₁, X₂,X₃,X₄ are independent N(0 ,1) variables, examine whether X₁+ 2X₂ -X₃ +3X₄ is

independent of  (X₁– X₂)² + (X₃ -X₄)² + (X₁ – X­₃)².

 

 

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

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

1. b) Let X₁, X₂, …,X_n denote a random sample from lognormal distribution. Show that

the sample geometric mean is lognormal.

20 a) Define trinomial distribution. State and establish its additive property.

1. b) If (X₁,X ₂)^/ is bivariate normal with correlation coefficient r , show that the correlation coefficient

between X₁² and X₂² is r².

21 a) Define non-central F distribution and derive its pdf .

1. b) Discuss any two applications of non-central F distribution.

22 a) Let X₁, X₂, …,X_n be independent N(0,1) variables.State and establish a necessary and sufficient

condition for  X^/AX to be distributed as chi-square,where  X =(X₁,X₂,…,X_n)/ .

1. b) Explain compound distribution with an illustration

 

 

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