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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