Question45:-If *quadx* is a random variable with finite expectation, then the value of *quadxP(X<-x)* as ‘quadx->00° is
A:-infinity
B:-unity
C:-zero
D:-indeterminate
Correct Answer:- Option-C
Question46:-If *quadx* is a symmetric random variable with distribution function ‘quadF* and real valued characteristic

function ©, then for any ‘quadx* in R , quadF(x)="
A:- quadF(-x)*
8:- quadF(-x-0)°
C:-*quadF(-x-0)-1"
D:- quad 1-F(-x-0)*
Correct Answer:- Option-D

R

Question47:-If the characteristic function © of distribution function ‘quadF* is absolutely integrable on ~", then for
SR
any’ quadx* in ೫ , quad f'={dF(x)}/dx" is
A:-bounded

B:-uniformly continuous

C:-both (1) and (2)

D:-Neither (1) nor (2)

Correct Answer:- Option-C
Question48:-Let ‘quadX* and ‘quadX_n* be independent standard normal variables on a probability space (Q, quadfrA,P)* *
*, for *quadn>=1* . Then which of the following is not true?

A:-X_nstackrel(P)(->)X*

B:-X_nstackrel(d)(->)X*

C:-quadE(X_n-X)=0°

D:- quadVar(X_n-X)=2°

Correct Answer:- Option-A
Question49:-The sequence *quad{X_n}° of independent random variables, each with finite second moment, obeys SLLN if

A:-quadsum_(k=1)“ooVar(X_k)
B:-quadsum_(k=1)“oo{Var(X_k)}/k
C:-‘quadsum_(k=1)*00{Var(X_k)}/sqrt(k)<0o*

D:- quadsum_(k=1)*o0{Var(X_k)}/k*2
Correct Answer:- Option-D
Question50:-Let ‘quad {X_n}* sequence of independent random variables with
*quadP(X_k=+-k)=1/2k*-Lambda* and ‘quadP(X_k=0)=1-k*-Lambda® , for ‘quadk>=1*
Then the sequence does not obey CLT if

A:-quadLambda=0"

B:-‘quadLambda=1*

C:-quadLambdain(0,1/2)*

0:- quadLambdain(1/2,1)*

Correct Answer:- Option-B
Question51:-Let ‘quadX* be a random variable with probability mass function

‘quad p(x) = {((6)pi*2 x*2) for x=1 ; -2 ; 3; -4 ...),(0 elsewhere):}°

Then
A:-quadE(X)=00°
B:-‘quadE(X)* exists
C:-‘quadE(X)D:- quadE(X)
Correct Answer:- Option-D
Question52:-Let ‘quad(X,Y)° has joint density
* quadf(x,y)={(1/8(6-x-y) 0<=x<2; 2<=y<4),(0 "elsewhere"):}>
Then ‘quadP(X+Y<3)="