Madurai Kamraj University (MKU) 2007 M.Sc Mathematics Mathematical Statistics - Question Paper

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n

4. Find the mgf, mean and variance of gamma distribution.

continuous random variable, find the pdf of 8X3 .

6.    If Y₁,Y₂,Y₃ is the order statistics of a random sample of size 3 from a distribution having pdf

/(*) = j^{1 0<x<1} find the pdf of Z_x = Y_Z -Y₁.

[0, elsewhere,

7.    Find the confidence intervals for means with known variance a².

8.    If S² is the variance of a random sample of size n > 1 from a distribution that is n(/u, 0),O < 0 < , what

nS2

is the efficiency of the estimator-?

n -1

SECTION B (3 x 20 = 60 marks)

Answer any THREE questions.

9.    (a) State and prove any four properties of a distribution function.

<²

(b) If X has the mgf M (t) = e², - <t < , find the expectation of all powers ofX

10.    If X and Y are continuous random variables with joint pdf fix, y),

(a)    find the conditional mean of Y, given X - x (if it is linear)

(b)    variance of the conditional distribution.

\

11.    If X_x and X₂ are two samples froma distribution

u    \ f¹- ^0<<1    I

with pdf fix) = <    *

[0, elsewhere,

(a)    find the joint pdf of Y₁ = X_t + X₂ and

Y₂= x,-x₂

(b)    find the marginal pdf of Y_r and Y₂

12.    (a) Derive Student's -distribution.

(b) If Z_n is j²(/i), prove that Y_n =iZ_n - n)/*j2n has a limiting normal distribution with mean 0 and variance 1.

13.    (a) State and prove Neyman-Pearson theorem.'

(b) If X is bin, p) and Y =    , prove that

Vnpq

Y² is x² (1) approximately.

14.    (a) State and prove Rao-Cramer inequality.

(b) If X₁,X₂, ,X_n is a random sample from a Poisson distribution with mean 6 > 0 , find an efficient estimator of Q.

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