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.

 

 

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        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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