Register Number :

Name of the Candidate :

1 9 8 0

M.Sc. DEGREE EXAMINATION, 2010

( MATHEMATICS )

( SECOND YEAR )

( PAPER - VIII )

240. MATHEMATICAL STATISTICS

( Including Lateral Entry )

May ] [ Time : 3 Hours

Maximum : 100 Marks

Turn over

SECTION - A (8 5 = 40)

Answer any EIGHT questions.All questions carry equal marks.

1. State and prove the multiplication rule ofprobability.

2. Let X be a random variable on a probabilityspace. Let E(1 1k) < for some k > 0.Then, prove that

nk P {1 1 > n } 0 as n .

12. (a) Explain the negative binomial distribution.

(b) Define Gamma distribution. For Gammadistribution, find E(X), var (X) and E(Xn).

13. (a) State and prove the Kolmogorvosinequality.

(b) With the usual notation, prove that

1 R1(23) = (1 12) (1 132).14. Prove that the random vectors ( x, y ) and

(x1 x, x2 x, , xn x, y1 y, , yn y)

are independent and the joint distribution of( x, y ) is a bivariate normal distribution withparameter 1, 2, , 12/n, 22/n.

15. (a) State and prove the factorization criterionfor sufficient statistic.

(b) State and prove a necessary and sufficientcondition for an unbiased estimate to bea uniformly minimum variance unbiasedestimate.

4

2 2 2

3. Suppose X is a random variable which have aGamma distribution, then find E(X) and Var (X).

4. Explain the properties of the correlationco-efficients.

5. In a trivariate population1 = 3, 2 = 4, 3 = 5, 23 = 04, 31 = 06and 21 = 07.Find :

(i) 231and (ii)

123.

6. Let { Xn } be any sequence of randomvariables. Let

Prove that a necessary and sufficient conditionfor the sequence { Xn } satisfy the weak lawof large numbers is that

7. Let X1, X2, , Xn be a sample from apopulation with distribution function F. Then,prove that

2 3

Yn = n1

n

k = 1Xk.

8. If X ~ F(m, n). For integer k > 0, show that

9. Define when an estimate is said to be

(i) Location invariant.

(ii) Scale invariant.

(iii) Location and scale invariant.

(iv) Permutation invariant.

10. Explain the terms minimal sufficient partitionand minimal sufficient statistic.

SECTION - B (3 20 = 60)

Answer any THREE questions.All questions carry equal marks.

11. Let X be a non - negative random variablewith distribution function F. Then prove that

E ( X ) = and var ( X ) =2n .

E(Xk) =hm( )

km2( )k + m2( ) k ][ ] [m2( )][ h2( )][

for h > 2k.

Turn over

Yn2

1 + Yn2{ } 0 as n .E

E(X) = 0

[ 1 F(x) ] dx.