Loyola College M.Sc. Statistics April 2008 Advanced Distribution Theory Question Paper PDF Download

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.

  1. 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)

 

  1. 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.

  1. State and establish the additive property satisfied by bivariate binomial distribution.
  2. Show that the regression equations are linear in the case of bivariate Poisson distribution.
  3. For lognormal distribution, show that mean > median > mode.
  4. State the mgf of Inverse Gaussian distribution. Hence find the cumulants.
  5. State and establish any two properties of Pareto distribution.
  6. Let X1, X2, X3 be independent standard normal variables. Find the mgf of

X12 + 5X22 + 4X32  – 2 X1X2 + 8X2X3 .

  1. 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 + Xand X1 – X2 are independent if and only if X1 is normal.

  1. b) Let X1,X2,…,Xn be independent random variables such that Xi has Inverse Gaussian distribution with

parameters i, µi , I = 1,2,…,n. Show that

(Xi – µi)2 / (Xi µi2)  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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