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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