BA 21

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – November 2008

    ST 1810 – ADVANCED DISTRIBUTION THEORY

 

 

 

Date : 08-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

SECTION – A                                                                       

Answer ALL the questions                                                                                        (10 x 2 = 20 marks)

 

1. Define a truncated distribution and give an example.
2. Find the MGF of a power series distribution.
3. Define lack of memory property for discrete random variable.
4. If X is distributed as Lognormal, show that its reciprocal is also distributed as Lognormal.
5. Let (X₁, X₂) have a bivariate Bernoulli distribution. Find the distribution of X₁ +  X₂.
6. Find the marginal distributions associated with bivariate Poisson distribution.
7. Show that Marshall – Olkin bivariate exponential distribution satisfies bivariate lack of memory property.
8. Define non-central chisquare – distribution and find its mean.
9. Let X_1, X₂, X₃, X₄ be independent standard normal variables. Examine whether

X₁² +3 X₂² + X₃² +4 X₄² – 2 X₁X₂ + 6 X₁X₃ + 6 X₂X₄– 4 X₃X₄  is distributed as chi-square.

10. Let X be N(q, 1), q = 0.1, 0.5. If q is discrete uniform, find the mean of the compound

distribution.

 

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

 

11. State and establish a characterization of Poisson distribution.
12. Derive the pdf of a bivariate binomial distribution. Hence, show that the regressions are

linear.

13. Let (X₁, X₂) follow a Bivariate normal distribution with V(X₁) = V(X₂). Examine

whether  X₁ + X₂  and (X₁ – X₂)²  are independent.

14. Show that the mean of iid Inverse Gaussian random variables is also Inverse Gaussian.
15. Let (X_1, X₂) follow a Bivariate exponential distribution . Derive the distributions of Min{X₁, X₂} and Max{X₁, X₂}.
16. Find the mean and the variance of a non-central F – distribution.
17. Let X_1, X₂, X₃,…, X_n be iid N(0, σ²), σ > 0 random variables.Find the MGF of X ^/AX/ σ².

Hence find the distribution of X₁X₂.

18. Illustrate the importance of the theory of quadratic forms in normal variables in ANOVA.

 

 

 

SECTION – C

Answer any TWO questions                                                                                        (2 x 20 = 40 marks)

 

19. a) Let X_1, X₂, X₃,…, X_n be iid non-negative integer-valued random variables. Show that X₁

is geometric if and only if Min{X_1, X₂, X₃,…, X_n} is geometric.

 

1.  b) State and establish the additive property of  bivariate Poisson distribution.

 

20. a) Let (X_1, X₂) have a bivariate exponential distribution of Marshall-Olkin. Find the

cov(X_1, X₂).

 

1.  b) Let (X_1, X₂) follow a bivariate normal distribution. State and establish any two of its

properties.

 

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

 

1.  b) Let X  be a random variable with the distribution function F given by

0 ,                 x < 0

F(x) =   (2x + 1)/4,    0  ≤ x < 1

1,                   x  ³ 1.

Find the mean, median and variance of X.

 

 

22. a) State and establish a necessary and sufficient condition for a quadratic form in normal variables to

have a chi-square distribution.

 

2. b) Let (X_1, X₂) follow a trinomial  distribution with index n and cell probabilities θ₁ ,θ₂. If the prior

distribution is uniform, find the compound distribution. Hence find the means of  X₁ and X₂.

 

 

Go To Main page

Latest Govt Job & Exam Updates:

View Full List ...