Question45:-If *quadX’ is a random variable with finite expectation, then the value of *quadxP(X<-x)" as "quadx->00" 5

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 805095
கயா
B:-*quadF(-x-0)"
C:-*quadF(-x-0}-1"
D:-"quad1-F(x-0}
Correct Answer:- Option-D

R

Question47:-If the characteristic function ® of distribution function “quadF" is absolutely integrable on =", then for

R
any'quadx’ in =2 ", “quad f'={dF(x)}/dx ‏ىا‎
‎கம்லா
‎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)}/kC:-*quadsum_{k=1)"o0{Var(X_k)}/sqri(k}D:-"quadsum_{k=1}"oo{Var(X_k}}/k" 2<00"
Correct Answer:- Option-D
Question50:-Let *quad {X_n}" sequence of independent random variables with
“quadP(X_k=+-k)=1/2k~-Llambda’ and ‘quadP(X_k=0)=1-k~-Lambda’ , for *quadk>=1"
Then the sequence does not obey CLT if
A:-"quadLambda=0"
B:-'‘quadlambda=
C:-"quadLambdain(0,1/2)"
D:-*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)}forx=1 ; -2 ; 3; -4 ..}{0elsewhere):}*

Then
A:-"quadE(X)=00"
B:-"quadE(X)" exists
C:-"quadE{X}<00" and “quadE(X)’ exists
D:-*quadE{X)Correct Answer:- Option-D
Question52:-Let “quad(X.Y)" has joint density
“quadfix,y)={(1/8(6-x-y} O<=x<2; 2<=y<4},(0 "elsewhere"):}"
Then ` १५५०५१८३} =`