LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 27

FIRST SEMESTER – APRIL 2006

                                          ST 1810 – ADVANCED DISTRIBUTION THEORY

(Also equivalent to ST 1806/ST 1803)

 

 

Date & Time : 20-04-2006/AFTERNOON   Dept. No.                                                       Max. : 100 Marks

 

 

Section – A (2×10 = 20 marks)

Answer ALL the questions

1. If X and Y are independent Binomial variates with same parameters (n, p), show that the conditional distribution of X given by X+Y is a Hyper geometric distribution.
2. Let X_n be discrete uniform on {1/n, 2/n, 3/n …1}, n Є N. Find the moment generating function (MGF) of X_n.
3. Define truncated Poisson distribution, truncated at zero and hence find its mean.
4. State and prove the additive property of bivariate Binomial distribution.
5. Show that for a random sample of size 2 from N(0, σ²)  population, E[X₍₂₎] = σ/√п
6. If (X₁, X₂) is bivariate normal, show that (X₁-X₂) is normal.
7. Define bivariate exponential distribution.
8. Show that in the case of bivariate exponential distribution, marginal distributions are exponential.
9. Write down the density function of non-central t-distribution. What is its non-centrality parameter?
10. Find the mean of non-central χ²– distribution.

Section – B (5×8 = 40 marks)

Answer any FIVE questions

1. Find the MGF of power series distribution. Show that Binomial and Poisson distributions are particular cases of power series distribution.
2. Establish the recurrence relation satisfied by raw moments of log-series distribution. Hence or otherwise, obtain the mean and variance of log-series distribution.
3. In a trinomial distribution with the parameters (n, p₁, p₂), show that the marginal distributions are Binomial. Also, find the correlation coefficient between X₁ and X₂.
4. If (X₁, X₂) is bivariate Poisson, obtain the conditional distributions and the regression equations.
5. For lognormal distribution, show that mean > median > mode.

 

 

 

 

 

 

 

1. Let X₁ and X₂ be independent and identically distributed random variables with positive variance. If (X₁ +X₂) and (X₁-X₂) are independent, show that X₁ is normal.
2. Show that the ratio of two independent standard normal variates is a Cauchy variate. Is the converse true?
3. State and prove the additive property of Inverse Gaussian (IG) distribution.

Section- C

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

19.a.  Show that in the case of multinomial distribution, multiple regressions are     linear. Hence find the partial correlation coefficient.         (10 marks)

1. State and establish the additive property of trinomial distribution. (10 marks)

20.a. Obtain the MGF of bivariate Poisson distribution with the parameters (λ₁, λ₂, λ₃). Also find the covariance of bivariate Poisson distribution.          (10 marks)

1. Let (X₁, X₂₎ be bivariate Poisson. Find the necessary and sufficient condition for X₁ and X₂ to be independent. (10 marks)

 

21.a.  Let X₁, X₂ X₃, X₄  be independent  N(0,1) random variables. Find the distribution of (X₁X₄ – X₂ X₃)      (10 marks)

22. Let (X₁, X₂) have bivariate normal distribution with the parameters (0,0,1,1,ρ) . Find the correlation coefficient between X₁² and X₂².    (10 marks)

22.a.  Derive the density function of non-central F-distribution. (10 marks)

1. Find the mean and variance of non-central F-distribution.(10 marks)

 

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                        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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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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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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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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