Sunday, 26 March 2017

HSST Statistics Solved Paper SR For SC/ST 16/2017/OL - Part 5

81.       The Poisson process with parameter λ is a renewal counting process for which the unit lifetimes have ................ distribution with common parameter λ.
(A) Poisson
(B) Exponential
(C) Uniform
(D) Geometric
Answer: B
82.       Let `{X_n, n = 0, 1, 2...}` be a Branching process and the corresponding offspring distribution has a pgf `P (s)=(2)/(3) +(s+s^2)/(6) ` . Find the probability of extinction of the process
(A) 0
(B) 0.25
(C) 0.66
(D) 1
Answer: D
83.       Let `{X_n}` be a renewal process with `mu = E (X_1) <oo ` and if `M (t)` is the renewal function, then `lim_(t->oo) (M (t))/(t) = .....`
(A) `(1)/(mu) `
(B) `mu `
(C) `(t)/(mu) `
(D) `(mu)/(t) `
Answer: A
84.       If `X_i`'s are independent Poisson variates with respective parameters `lambda_i`, for `i = 1, 2, ...k`, then the conditional distribution of `X_1, X_2,... X_k` given their sum `sum_(i=1)^k X_i = n` is a ................. distribution with parameters ....................... and ......................
(A) Binomial with parameters `n` and ` (1)/(k)`
(B) Binomial with parameters `k` and `(1)/(n)`
(C) Multinomial with parameters `n` and `(1)/(k)`
(D) Multinomial with parameters `k` and `(1)/(n)`
Answer: C
85.       If ` (X_1, X_2)` is a Bivariate normal random vector with parameters `(mu_{X1}, mu_{X2}, sigma ^2_X_1,sigma^2_X_2, rho`), when `sigma^2_X_1 = sigma^2_X_2 ` and `rho = 0` , the density function is called
(A) Elliptical Normal
(B) Circular Normal
(C) Symmetrical Normal
(D) Uniform Normal
Answer: B
86.       If the random vector `X` follows Multivariate Normal distribution with mean vector 0 and dispersion matrix `I` and `Q_i = X^' A_i X` are quadratic forms of rank `r_i` such that `sum_(i=1)^k A_i = I_p` , then a necessary and sufficient condition for `Q_i`'s to be distributed as independent chi-square random variables with `r_i` d.f is that
(A) ` sum_(i=1)^k r_i=k`
(B) `sum_(i=1)^k r_i = p`
(C) `sum_(i=1)^k r_i=0`
(D) `sum_(i=1)^k r_i = kp`
Answer: B
87.       The relationship between partial correlation coefficients `r_{ij.k},` multiple correlation
coefficients `R_{i.jk}`and simple correlation coefficients `r_{ij}` is
(A) `R^2_1.23 = 1+ (1-r^2_12) (1 - r^2_13.2)`
(B) `R^2_1.23 = 1 - (1- r^2_12) (1- r^2_13.2)`
(C) `R^2_1.23 = 1 + (1-r^2_12)// (1-r^2_13.2)`
(D) `R^2_1.23 = 1- (1-r^2_12) // (1-r^2_13.2)`
Answer: B
88.       Hotelling's `T^2` statistic and Mahalnobis `D^2` statistic are connected by the relationship
(A) `D^2 = ((N_1 N_2))/((N_1+N_2)) T^2`
(B) `D^2 =((N_1 N_2))/((N_1-N_2)) T^2`
(C) `D^2 = ((N_1-N_2))/((N_1 N_2)) T^2`
(D) `D^2 = ((N_1 + N_2))/((N_1 N_2)) T^2`
Answer: D
89.       In principal component analysis the variances of the Principal Components are the .................... of the covariance matrix.
(A) diagonal elements
(B) eigen values
(C) normalized elements
(D) non-zero elements
Answer: B
90.    For discriminating between two populations R.A. Fisher suggested the linear discriminant function `X'l` for which
(A) ` ("(mean difference)"^2)/("variance")`
(B) `("mean difference")^2/("A.M.")`
(C) `("mean difference")/("median")`
(D) `("variance")/("mean difference")`
Answer: A
91.    Assume that the time to failure `(T)` for a certain bulb has an exponential distribution `f ((t)/(lambda))` with parameter `lambda >0` with the prior pdf `g (lambda)` of `lambda` is an exponential distribution with parameter 2. Then the posterior pdf of `lambda` given `T = t` is
(A) `(2)/(t+2)`
(B) `(lambda)/(e^lambda (t+2))`
(C) `(lambda e^(lambda (t+2)))/((t+2)^2)`
(D) `(lambda (t+2)^2)/(e^(lambda (t+2)))`
Answer: D
92.    The basic elements of statistical decision theory is
(A) a space Ω `= {ul theta}` of all possible states of nature
(B) an action space `A = {a}` of all possible courses of action
(C) a loss function `L (ul theta, a)` giving the incurred loss when action `a` is taken and the state is ` ul theta`
(D) all these
Answer: D
93.    When there is no censoring for the life length `T`, the general formula of a survival function is
(A) `hat {S (t)} = (" # of individuals with" T >= t)/("total sample size")`
(B) `hat {S (t)} = (" # of individuals with" T <= t)/("total sample size")`
(C) `hat {S (t)} = (" # of individuals with" T = t)/("total sample size")`
(D) `hat {S (t)} = (" # of individuals with" T = 0)/("total sample size")`
Answer: A
94.    The Cox's Proportional Hazard Model (Cox's PH Model) with explanatory variables ` ul X = (X_1, X_2, ... X_p),beta_i` their regression coefficients and `h_0 (t)` a base line hazard, is `h (t, ul X) =`
(A) `e^{h_0 (t) sum_(i=1)^p beta_i X_i}`
(B) `log h_0 (t) + sum_(i=1)^p beta_i X_i`
(C) `h_0 (t) e sum_(i=1)^p beta_i X_i`
(D) `e^(h_0 (t)) sum_(i=1)^p beta_i X_i`
Answer: C
95.    When an inspection lot contains no defectives the OC function `L (p)` is
(A) `L (p) = 1`
(B) `L (p) = oo`
(C) ` L (p) = 0`
(D) None of these
Answer: A
96.    In a Time series data, the two main components which cause lack of stationarity are
(A) Seasonal and irregular variations
(B) Cyclic and irregular variations
(C) Trend and cyclic variations
(D) Trend and seasonal variations
Answer: D
97.    In the ARMA (1, 1) model `Z_t = ``phi Z_{t-1} + epsilon _t - theta epsilon_{t-1}` the condition for stationarity and invertilibility are respectively
(A) ` | phi | <= 1 and | theta | < 1` with `phi != theta`
(B) ` | phi | <= 1 and | theta | < 1` with `phi = theta`
(C) ` | phi | > 1 and | theta | > 1` with `phi!= theta`
(D) ` | phi | > 1 and | theta | > 1` with `phi = theta`
Answer: A
98.    In a Linear programming Problem with `n + m` variables and `m` constraints the number of basic solutions is
(A) ` ((n+m),(m))`
(B) `((n),(m))`
(C) `((m),(n))`
(D) `((n+m),(n-m))`
Answer: A
99.    If the demand curve is of the form `p = ae^{-bx}` , where `p` is the price and `x` is the demand, then the price elasticity of demand is
(A) `eta_p = bx`
(B) `eta_p = - bx`
(C) `eta_p = 1//bx`
(D) `eta_p = - 1 // bx`
Answer: C
100.    The Engel's curves for constant prices and those for constant incomes are respectively
(A) Concave and Convex
(B) Convex and Concave
(C) Both Concave
(D) Both Convex
Answer: B

Saturday, 25 March 2017

HSST Statistics Solved Paper SR For SC/ST 16/2017/OL - Part 4

61.       "Simple random sampling" is the technique of drawing a sample in such a way that each unit of the population has
(A) distinct and dependent chance of being included in the sample
(B) distinct but independent chance of being included in the sample
(C) an equal but dependent chance of being included in the sample
(D) an equal and independent chance of being included in the sample
Answer: D
62.       In SRSWR with usual notations, the standard error of the sample mean `quadbary` is
(A) `quadS({N-n}/{Nn})^{1/2}`
(B) `quadS({N-1}/{Nn})^{1/2}`
(C) `quadS(1-{n}/{N})^{1/2}`
(D) `quadS/n(1-{1}/{N})^{1/2}`
Answer: B
63.       The formulae for optimum allocation in various strata in stratified sampling were first derived by
(A) Tschuprov
(B) Cochran
(C) Lahiri
(D) Neymann
Answer: A
64.       The ratio estimator of population mean is unbiased if sampling is done according to
(A) PPSWR
(B) PPSWOR
(C) SRSWR
(D) SRSWOR
Answer: A
65.       The cluster sampling is more efficient when
(A) the variation within clusters in more
(B) the variation between clusters is less
(C) both (A) and (B)
(D) neither (A) nor (B)
Answer: C
66.       Local control is a device to maintain
(A) homogeneity within blocks
(B) homogeneity among blocks
(C) both (A) and (B)
(D) neither (A) nor (B)
Answer: A
67.       In a linear model `quadY_{ij}=alpha_i+e_{ij},` `quadj=1,2,...,n_i;` `quadi=1,2,...,k,` consider
(i) `quadalpha_1-3alpha_2+alpha_3+alpha_4`
(ii) `quadalpha_1+3alpha_2-alpha_3-alpha_4`
(iii) `quadalpha_1+3alpha_2-2alpha_3-2alpha_4`
Then which of the following is correct?
(A) (i) and (ii) are linear contrasts
(B) (i) and (iii) are linear contrasts
(C) (ii) and (iii) are linear contrasts
(D) (i), (ii) and (iii) are linear contrasts
Answer: B
68.       While analyzing the data of a `quadkxxk` Latin Square Design, the degrees of freedom in the ANOVA is
(A) `quadk^2-1`
(B) `quadk-1`
(C) `quadk^2-2k+1`
(D) `quad(k-1)(k-2)`
Answer: D
69.       In a split plot design with factor `quadA` at 3 levels in main plots, factor `quadB` at 3 levels in sub-plots and 3 replications, the degrees of freedom for sub-plot error is
(A) 27
(B) 12
(C) 8
(D) 4
Answer: B
70.    If the interactions `quadAB` and `quadBC` are confounded with incomplete blocks in a `quad2^n` factorial experiment, then automatically confounded effect is
(A) `quadA`
(B) `quadC`
(C) `quadAC`
(D) `quadABC`
Answer: C
71.    Which among the following is a consistent estimator of the population mean when samples are from the Cauchy population?
(A) Sample mean
(B) Sample median
(C) Sample variance
(D) None of these
Answer: B
72.    If the regularity conditions of the CR inequality are violated then the least attainable variance will be
(A) equal to the CR bound
(B) greater than the CR bound
(C) less than the CR bound
(D) zero
Answer: C
73.    A method to obtain the UMVUE is by using
(A) Rao-Blackwell Theorem
(B) Baye's Theorem
(C) Neymann-Pearson Theorem
(D) Lehmann-Scheffe Theorem
Answer: D
74.    A complete-sufficient statistic for `p` in the Bernoulli distribution
`(x, p) = p^x (1-p)^x; x=0, 1.
= 0 `"otherwise"` is
(A) The first order statistic `X_{(1)}`
(B) The `n` `"^{th}"` order statistic `X_{(n)}`
(C) `sum_(i=1)^n X_i`
(D) `X_{(n)}-X_{(1)}`
Answer: C
75.    The least square estimators are
(A) Unbiased
(B) BLUE
(C) UMVUE
(D) All these
Answer: D
76.    A 95% confidence interval for λ, when a large sample is taken from a Poisson population with parameter λ is
(A) `bar x ``+- 1.65``sqrt(( bar x)/(n)) `
(B) `lambda+- 1.65 sqrt((lambda)/(n)) `
(C) `bar x+- 1.96sqrt((bar x)/(n)) `
(D) `lambda +- 1.96sqrt((lambda)/(n)) `
Answer: C
77.    The minimum Chi-squared estimators are not necessarily
(A) Unbiased
(B) Consistent
(C) Efficient
(D) Asymptotically normal
Answer: A
78.    Which one of the following statements is true?
(A) Even if the UMP test does not exist, a UMPU test may exist
(B) Even if the UMPU test does not exist, a UMP test may exist
(C) A UMP test exists only if a UMPU test exists
(D) A UMPU test exists only if a UMP test exists
Answer: A
79.    In paired `t` test the two random variables should be
(A) Paired and uncorrelated
(B) Unpaired and correlated
(C) Both paired and correlated
(D) Neither paired nor correlated
Answer: C
80.    With usual notations, the criterion for acceptance in SPRT is
(A) `lambda_m <= ((1-beta))/(alpha) `
(B) `lambda_m >= ((1-beta))/(alpha) `
(C) `lambda_m <=(beta)/((1-alpha)) `
(D) ` lambda_m >= (beta)/((1-alpha)`
Answer: C

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