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Total No. of Pages :

Register Number : 6373

Name of the Candidate :

M.Sc. DEGREE EXAMINATION MAY 2014.

(MATHEMATICS)

(SECOND YEAR)

240 MATHEMATICAL STATISTICS

Time : Three hours Maximum : 100 marks

SECTION A

Answer any EIGHT questions. (8 5 = 40)

All questions carry equal marks.

1. State and prove the law of total probability.

2. If the distribution of X is symmetrical, then show that ( ) ( )xFxF =1 .

3. Let X be a gamma variate with parameter 1 and . Then prove that { } { }tXPsXstXP >=>+> | , where t and s are any two positive real

numbers.

4. If ( )4,3~NX , then find { }52 < XP (Given : ( ) 841.01 =ZP , ( ) 309.05.0 =ZP ).

5. In the usual notations, given that 31 = , 42 = , 53 = , 4.023 = , 6.031 = and 7.021 = . Calculate 1.23 and 23.1R .

6. If aXP

n and aYP

n (a and b are constants), then prove that

abYXP

nn .

7. Show that ( )

2

21

Sn is a chi-square variate with ( )1n degrees of freedom.

8. Sketch a two-way ANOVA table with one observation per cell.

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9. Let nXXX ,..., 21 be a sample from probability mass function

( )

=

=

otherwise.,0

,,...,2,1for,1

NKNKPN

Determine the maximum likelihood estimate N of N

10. Define

(a) Minimal sufficient partition U and

(b) Minimal sufficient statistic.

SECTION B

Answer any THREE questions. (3 20 = 60)

All questions carry equal marks.

11. (a) A urn contains 5 white balls and 7 red balls. Two balls are drawn in

succession. Find the probability that both the balls are white.

(b) State and prove Lyapunov inequality.

12. Find K so that ( ) Kxyyxf =, , ( )21 yx will be a probability density function. Also find the distribution function ( )YXF , , marginal probability density functions of X and Y .

13. State and prove Kolmogrov strong law of large numbers.

14. Explain two-way analysis of variance with one observation per cell.

15. (a) Let ( nXXX ,..., 21 ) be a sample from ( )1,N . Then prove that the statistic ( )

=

=

n

i

iXXT1

is minimal sufficient partition for .

(b) Find the Neyman-Pearson size test of 00 : =H against 11 : =H ( )21 < , based on a sample of size 1 from the probability density function ( ) ( )( )xxf += 1122 ( )10