Friday, 24 March 2017

HSST Statistics Solved Paper SR For SC/ST 16/2017/OL - Part 3

41.       If `quad{A_n}` is a sequence of events on a probability space (Ω,`quadA,P)` such that `quadA_n->A` as `quadn->oo` , then what is the value of lim`quadP(A_n)` ?
(A) zero
(B) one
(C) `quadP(A)`
(D) need not exist
Answer: C
42.       If `quadA` and `quadB` are mutually exclusive events, each with positive probabilities, then they are
(A) independent events
(B) dependent events
(C) equally likely events
(D) exhaustive events
Answer: B
43.       If `quad{A_n}` is a sequence of events such that `quadsum_(k=1)^ooP(A_k)=oo` , then `quadP(lim"sup"A_n)=1` provided events are
(A) equally likely
(B) Mutually exclusive
(C) independent
(D) pair-wise mutually exclusive
Answer: C
44.       Let `quad{A_n}` be a sequence of events such that `quadB_1=A_1` and `quadB_k=A^c_1 A^c_2...` `A_{k-1}^c A_k` for `quadk>=2` , in which `quadA^c` is the complement of `quadA` . Then the sequence of events `quad{B_n}` are
(A) Pair-wise independent
(B) Mutually independent
(C) Mutually dependent
(D) Pair-wise mutually exclusive
Answer: D
45.       If `quadX` is a random variable with finite expectation, then the value of `quadxP(X<-x)` as `quadx->oo` is
(A) infinity
(B) unity
(C) zero
(D) indeterminate
Answer: C
46.       If `quadX` is a symmetric random variable with distribution function `quadF` and real valued characteristic function Φ, then for any `quadx` in ,`quadF(x)=`
(A) `quadF(-x)`
(B) `quadF(-x-0)`
(C) `quadF(-x-0)-1`
(D) `quad1-F(-x-0)`
Answer: D
47.       If the characteristic function Φ of distribution function `quadF` is absolutely integrable on , then for any`quadx` in , `quad f'={dF(x)}/dx` is
(A) bounded
(B) uniformly continuous
(C) both (A) and (B)
(D) Neither (A) nor (B)
Answer: C
48.       Let `quadX` and `quadX_n` be independent standard normal variables on a probability space (Ω,`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`
Answer: A
49.       The sequence `quad{X_n}` of independent random variables, each with finite second moment, obeys SLLN if
(A) `quadsum_(k=1)^ooVar(X_k)<oo`
(B) `quadsum_(k=1)^oo{Var(X_k)}/k<oo`
(C) `quadsum_(k=1)^oo{Var(X_k)}/sqrt(k)<oo`
(D) `quadsum_(k=1)^oo{Var(X_k)}/k^2<oo`
Answer: D
50.    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)`
(D) `quadLambdain(1/2,1)`
Answer: B

51.    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)=oo`
(B) `quadE(X)` exists
(C) `quadE(X)<oo` and `quadE(X)` exists
(D) `quadE(X)<oo` , but `quadE(X)` does not exist
Answer: D
52.    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)=`
(A) `5/24`
(B) `5/8`
(C) `3/8`
(D) None of these
Answer: A
53.    If `quadX` and `quadY` are two random variables having finite expectations, then the value of `quadE["min"{X,Y}+"max"{X,Y}]` is
(A) less than `quadE(XY)`
(B) less than `quadE(X+Y)`
(C) equal to `quadE(XY)`
(D) equal to`quadE(X+Y)`
Answer: D
54.    The Poisson distribution `quadP(Lambda)` is unimodal when
(A) `quadlambda` is not an integer
(B) `quadlambda` is an integer
(C) Both (1) and (2)
(D) Neither (1) nor (2)
Answer: A
55.    Which of the following distribution is not a member of power series family of distributions?
(A) Binomial
(B) Poisson
(C) Geometric
(D) Hypergeometric
Answer: D
56.    If `quadX` follows normal `quadN(mu,sigma)` , then the approximate value of `quadE{|X-mu|}` is
(A) Zero
(B) `sigma`
(C) `quad4/5sigma`
(D) `quadsqrt(4/Pi)sigma`
Answer: C
57.    If `quadX` is uniformly distributed with mean unity and variance 0.75, then `quadP(X>1)=`
(A) 0.25
(B) 0.5
(C) 0.75
(D) 1
Answer: B
58.    If `quadX` follows normal `quadN(mu,Sigma)` , then `quadY=e^X` follows
(A) Log-normal distribution
(B) Exponential distribution
(C) Logistic distribution
(D) Pareto distribution
Answer: A
59.    If `quadX_j` follows exponential `quadE(Theta_j)` distribution, for `quadj=1,2,...,n,` then the distribution of `quad"min"{X_1,X_2,...,X_n}`
(A) `quadE(Theta_j)`
(B) ` E (prod_{j=1}^n theta_j)`
(C) `quadE(sum_(j=1)^nTheta_j)`
(D) `quadE["min"{Theta_1,Theta_2,...,Theta_n}]`
Answer: C
60.    The mode of `quadF` -distribution is
(A) always less than unity
(B) sometimes less than unity
(C) always greater than unity
(D) sometimes equal to unity
Answer: A

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