Download DUET Master 2018 DU MPhil PhD in Statistics Question Paper With Answer Key

Download DUET (Delhi University Entrance Test conducted by the NTA) 2018 DU MPhil PhD in Statistics Question Paper With Solution Key

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DUET Last 10 Years 2011-2021 Question Papers With Answer Key || Delhi University Entrance Test conducted by the NTA

DU MPhil PhD in Statistics

Topic:- DU_J18_MPHIL_STATS_Topic01

Which of the following is a point of PG(3, 2)? [Question ID = 17310]
1. (0, 2, 0) [Option ID = 39232]
2. (0, 1, 1, 1) [Option ID = 39233]
3. (0, 0, 0, 0) [Option ID = 39234]
4. (1, 0, 1) [Option ID = 39231]
Correct Answer :-
- (0, 1, 1, 1) [Option ID = 39233]
Suppose a cricket ball manufacturing company formed lots of 500 balls. To check the quality of the lots, the buyer draws 20 balls from each lot and accepts the lot if the sample contains at the most 1 defective ball. If the quality of submitted lot is 0.03, the AOQ for this plan under corrective sampling is [Question ID = 17248]
1. 0.015 [Option ID = 38983]
2. 0.035 [Option ID = 38985]
3. 0.025 [Option ID = 38984]
4. 0.045 [Option ID = 38986]
Correct Answer :-
- 0.025 [Option ID = 38984]
In time series analysis, the Box-Jenkins method is based on fitting which of the following models: [Question ID = 17242]
1. autoregressive moving average (ARMA) [Option ID = 38959]
2. autoregressive integrated moving average (ARIMA) [Option ID = 38960]
3. none of these [Option ID = 38962]
4. both autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) [Option ID = 38961]
Correct Answer :-
- both autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) [Option ID = 38961]

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The number of points in a 3-dimensional subspace of PG (4, 2) is: [Question ID = 17227]
1. 15 [Option ID = 38902]
2. 31 [Option ID = 38899]
3. 13 [Option ID = 38901]
4. 21 [Option ID = 38900]
Correct Answer :-
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- 15 [Option ID = 38902]
If {Xn=1} is a sequence of independent and identically distributed standard Cauchy variates and Z = (X1 + X2 +...+Xn)/n, then the value of E(Z²/(1+Z²)) is: [Question ID = 17309]
1. 1 [Option ID = 39227]
2. 0 [Option ID = 39230]
3. 0.25 [Option ID = 39229]
4. 0.5 [Option ID = 39228]
Correct Answer :-
- 0.5 [Option ID = 39228]
Consider the problem of testing Ho: X~ Normal with mean 0 and variance 1/2 against H1: X ~ Cauchy (0, 1). Then for testing Ho against H1, the most powerful size a test [Question ID = 17257]
1. rejects Ho if and only if |x| > c2 where c2 is such that the test is of size a [Option ID = 39020]
2. rejects Ho if and only if x < C4 or x > C5, where c4 < c5 where c4 and c5 are such that the test is of size a [Option ID = 39022]
3. rejects Ho if and only if |x| < c3 where c3 is such that the test is of size a [Option ID = 39021]
4. does not exist [Option ID = 39019]
Correct Answer :-
- rejects Ho if and only if |x| > c2 where c2 is such that the test is of size a [Option ID = 39020]
If X is a p- component random vector with V(X) = ? and if A is any constant matrix of order p x p, then V (AX) is equal to [Question ID = 17233]
1. A'?A [Option ID = 38924]
2. ? [Option ID = 38926]
3. A? [Option ID = 38925]
4. A?A' [Option ID = 38923]
Correct Answer :-
- A?A' [Option ID = 38923]
Let X1, X2,...,Xn be a random sample from a population with pmf P?(X = x) = ?x(1 - ?)1-x, x = 0 or 1 and 0 = ? = 1/2 Then the MLE of ? is: [Question ID = 17253]
1. ?(i=1 to n)Xi/n [Option ID = 39003]
2. X/2 [Option ID = 39004]
3. Min{Xi} [Option ID = 39005]
4. 1/2 [Option ID = 39006]
Correct Answer :-
- Min{Xi} [Option ID = 39005]

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Consider a renewal process {Nt; t= 0} for which the inter arrival time follows U(0,1) distribution. The renewal function for 0 = t = 1 is given by [Question ID = 17223]
1. e^(t-1) [Option ID = 38885]
2. 2t-1 [Option ID = 38886]
3. e^(-2t-1) [Option ID = 38883]
4. e^(-2t) [Option ID = 38884]
Correct Answer :-
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- e^(t-1) [Option ID = 38885]
For a two-way random effects model with m observations per cell, factor A has p levels and factor B has q levels. Then the expression for sl² is: [Question ID = 17229]
1. (MSA-MSB)/pq [Option ID = 38909]
2. None of these [Option ID = 38910]
3. (MSB-MS(AB))/pm [Option ID = 38908]
4. (MSA-MS(AB))/qm [Option ID = 38907]
Correct Answer :-
- (MSA-MS(AB))/qm [Option ID = 38907]
Let X1~?(µ, s²) and X2, X3, Xn be a sample of size (n-1) drawn from N(0,s²). Further, let X1 is independent of X2, X3, Xn. Then ?²= (X2²+X3²+...+Xn²)/s² has a chi- square distribution with
1. (n-1) degrees of freedom
2. n degrees of freedom
3. non centrality parameter µ²/s²
4. non centrality parameter nµ²/s²
Which of the above is/are correct? [Question ID = 17216]
1. 1 only [Option ID = 38855]
2. 2 only [Option ID = 38856]
3. Both 2 and 3 [Option ID = 38857]
4. Both 1 and 4 [Option ID = 38858]
Correct Answer :-
- Both 2 and 3 [Option ID = 38857]

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Let {Xn=1} be a sequence of random variables, with Cov.(Xi,Xj) = { 1, i = j; 1/2, |i- j| = 1,2,3; 0, otherwise. } Then the value of Var.(X1+X2 +...+Xn)/n² is: [Question ID = 17208]
1. (4-6n)/n² [Option ID = 38823]
2. (4n-6)/n² [Option ID = 38825]
3. (4n+6)/n² [Option ID = 38824]
4. (4/n - 6/n²) [Option ID = 38826]
Correct Answer :-
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- (4/n - 6/n²) [Option ID = 38826]
If A = ((A11, S11, S12), (A12, S21, S22)) ~ Wp (n, S), where S = ((S11, S12), (S21, S22)), then, with usual notations, the distribution of A22.1 = A22-A21A11A12 is [Question ID = 17234]
1. Wx(n - p + k, 222.1) [Option ID = 38929]
2. Wx(n-k, 222.1) [Option ID = 38930]
3. Wp-k(? - ?, S22.1) [Option ID = 38928]
4. Wp-k(n - p + k, S22.1) [Option ID = 38927]
Correct Answer :-
- Wp-k(? - ?, S22.1) [Option ID = 38928]
A sample of size n (n = 2) is drawn from a finite population of N units by probabilities proportional to size sampling with selection probability pi; (1= i = N, 0 < pi < 1, S(i=1 to N)Pi=1). Let T = S(i=1 to n) yi/pi where yi is the value of a study variable for the ith unit and the sum extends over the units included in the sample. Which of the following statements is true? [Question ID = 17244]
1. the variance of T reduces to 0 if pi = 1/N ; for all i; 1= i = N. [Option ID = 38969]
2. Variance of T remains same. [Option ID = 38970]
3. Tis an unbiased estimator of the population total S(i=1 to N) Vi [Option ID = 38967]
4. nT is an unbiased estimator of the population total S(i=1 to N) Vi [Option ID = 38968]
Correct Answer :-
- Tis an unbiased estimator of the population total S(i=1 to N) Vi [Option ID = 38967]
If X ~ ?p (µ, S ), then the distribution of X'S?¹X is [Question ID = 17232]
1. Central Chi-square [Option ID = 38921]
2. Multivariate normal [Option ID = 38920]
3. Univariate normal [Option ID = 38919]
4. Non-central Chi-square [Option ID = 38922]
Correct Answer :-
- Non-central Chi-square [Option ID = 38922]
Let Xa(a = 1,2,..., N) be N independent observations from Np(µ, S), X = (1/N)S(a=1 to N) Xa and let S be the maximum likelihood estimator for S. Then which one of the following statements is true? [Question ID = 17236]
1. X and (2/N)S(a=1 to N)(Xa - X) (Xa - X)' are independently distributed. [Option ID = 38936]
2. S and (2/N)S(a=1 to N)(Xa – X) (Xa – X)' are independently distributed. [Option ID = 38937]
3. None of these. [Option ID = 38938]
4. S is an unbiased estimator for S. [Option ID = 38935]
Correct Answer :-
- X and (2/N)S(a=1 to N)(Xa - X) (Xa - X)' are independently distributed. [Option ID = 38936]

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According to Slutsky's theorem if Xn -d->X, Zn -p->Z and Yn -p->a, where 'd' stands for distribution and 'p' stands for probability, then
1. YnXn -d->aX
2. YnXn -p->aX
3. Xn+Yn -d->X+a
4. Xn+Zn -d->X+Z
Which of the above is/are correct? [Question ID = 17217]
1. Both 1 and 3 [Option ID = 38861]
2. Both 2 and 4 [Option ID = 38862]
3. 2 only [Option ID = 38860]
4. 4 only [Option ID = 38859]
Correct Answer :-
- Both 1 and 3 [Option ID = 38861]
Let X and A denote, respectively, the sample mean vector and the Wishart matrix obtained from a random sample Xa(a = 1,2,..., N) of size N drawn from Np (µ, S) and let y = N(X – µ)' A?¹(X – µ). Then (N-p).y/p is distributed as [Question ID = 17235]
1. Non-central Fp,N-p (?µ'S?¹µ) [Option ID = 38932]
2. Central Fp,N-p [Option ID = 38931]
3. Central ?² [Option ID = 38933]
4. Non-central ?² (?µ'S?¹µ) [Option ID = 38934]
Correct Answer :-
- Central Fp,N-p [Option ID = 38931]
The joint probability density function of two random variables X and Y is: f(x,y) = { 24xy, x > 0, y > 0 and x + y <1; 0, otherwise. } The value of E (Var (Y | X = x)) is: [Question ID = 17209]
1. 3/45 [Option ID = 38828]
2. 1/15 [Option ID = 38830]
3. 1/45 [Option ID = 38827]
4. 2/45 [Option ID = 38829]
Correct Answer :-
- 1/45 [Option ID = 38827]

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If X, Y, Z are independent and identically distributed standard uniform variates, then the value of E (1-2U)², where U = Max (X,Y,Z), is: [Question ID = 17213]
1. 2/5 [Option ID = 38844]
2. 1/3 [Option ID = 38843]
3. 3/5 [Option ID = 38846]
4. 1/3 [Option ID = 38845]
Correct Answer :-
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- 2/5 [Option ID = 38844]
Suppose X1, X2,... Xn is a random sample from U(0, ?), ? >0. Then [Question ID = 17256]
1. X(n) is sufficient statistics of ? [Option ID = 39017]
2. X(n) is not sufficient statistics of 0 [Option ID = 39015]
3. X(1) is sufficient statistics of ? [Option ID = 39016]
4. X(1) is MLE of ? [Option ID = 39018]
Correct Answer :-
- X(n) is sufficient statistics of ? [Option ID = 39017]
Suppose that a parallel system is composed of two identical components, each with failure rate ?=0.01. The system reliability for t=10 hours is [Question ID = 17252]
1. 0.89 [Option ID = 39001]
2. 0.79 [Option ID = 39000]
3. 0.69 [Option ID = 38999]
4. 0.99 [Option ID = 39002]
Correct Answer :-
- 0.99 [Option ID = 39002]
If v? (Yn-µ)~N(0,s²) and if 'g' is a differentiable function such that g'(µ) = 0 and g"(µ) exits and is not 0,then by Delta method n (g(Yn)-g(µ)) converges in distribution to [Question ID = 17219]
1. (s²g"(µ)/2) [Option ID = 38869]
2. (s²g"(µ)/?²) [Option ID = 38867]
3. (?²/2) [Option ID = 38868]
4. (s²g"(µ)) [Option ID = 38870]
Correct Answer :-
- (s²g"(µ)/?²) [Option ID = 38867]
Which one of the following distributions belong to the exponential family of distributions? [Question ID = 17214]
1. logistic distribution [Option ID = 38848]
2. Uniform (0,?) distribution [Option ID = 38847]
3. Beta distribution [Option ID = 38849]
4. Cauchy distribution [Option ID = 38850]
Correct Answer :-
- Beta distribution [Option ID = 38849]

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The approximate relationship between Durbin Watson d-statistic and sample first order correlation coefficient ? (an estimator of ?) is: [Question ID = 17240]
1. d˜1-? [Option ID = 38953]
2. d˜?/2 [Option ID = 38954]
3. d˜1-2? [Option ID = 38951]
4. d˜2(1-?) [Option ID = 38952]
Correct Answer :-
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- d˜2(1-?) [Option ID = 38952]
Starting at time t = 0, visitors enter a museum according to a Poisson process with rate 2. Each visitor spends a random time in the museum that is uniformly distributed between 0 and 1. Determine the expected number of visitors in the museum at time t = 5. [Question ID = 17225]
1. 5 [Option ID = 38893]
2. None of these [Option ID = 38894]
3. 1 [Option ID = 38892]
4. 2 [Option ID = 38891]
Correct Answer :-
- 1 [Option ID = 38892]
If immigrants arrive in a locality A at a Poisson rate of 10 per week and each immigrant is of African origin with probability 1/12, then find the probability that no person of African origin will emigrate to locality A during the month of February. [Question ID = 17222]
1. e^(-10) [Option ID = 38879]
2. e^(-10/3) [Option ID = 38881]
3. None of these [Option ID = 38882]
4. e^(-5/6) [Option ID = 38880]
Correct Answer :-
- e^(-10/3) [Option ID = 38881]
For any random variable X, let G (t) = E(t^X), then the value of lim(t->1) (1-G(t))/(1-t²) is: [Question ID = 17211]
1. E(X)/2 [Option ID = 38837]
2. -E(X) [Option ID = 38838]
3. E(X) [Option ID = 38835]
4. 2E(X) [Option ID = 38836]
Correct Answer :-
- E(X)/2 [Option ID = 38837]
The univariate analogue of Wishart distribution is [Question ID = 17237]
1. Exponential distribution [Option ID = 38939]
2. F-distribution [Option ID = 38941]
3. t - distribution [Option ID = 38940]
4. Chi-square distribution. [Option ID = 38942]
Correct Answer :-
- Chi-square distribution. [Option ID = 38942]

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To reduce the number of lags in Distributed lag models [Question ID = 17239]
1. Method of Instrumental Variables is used [Option ID = 38950]
2. Either Koyck's transformation or Almon's approach can be used [Option ID = 38949]
3. Almon's approach is used [Option ID = 38948]
4. Kocyk's transformation is used [Option ID = 38947]
Correct Answer :-
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- Kocyk's transformation is used [Option ID = 38947]
Which of the following processes are not second-order stationary: [Question ID = 17224]
1. Yn = Xn + a Xn-1 where{Xn: n= 1} is a sequence of independent and identically distributed random variables and a is a real constant. [Option ID = 38888]
2. {Xn: n= 1} be uncorrelated random variables with mean 0 and variance 1. [Option ID = 38889]
3. X1 = Z1 cos(?t) + Z2 sin(?t); ?t ? R and Z1, Z2 are independent normally distributed with mean 0 and variance s². [Option ID = 38887]
4. None of these [Option ID = 38890]
Correct Answer :-
- None of these [Option ID = 38890]
Method of Indirect Least Squares for estimation is appropriate when [Question ID = 17243]
1. None of these [Option ID = 38966]
2. simultaneous equations are unidentified [Option ID = 38965]
3. simultaneous equations are exactly identified [Option ID = 38964]
4. simultaneous equations are over identified [Option ID = 38963]
Correct Answer :-
- simultaneous equations are exactly identified [Option ID = 38964]
In ARIMA(p.d.q) model, p is: [Question ID = 17241]
1. the degree of differencing [Option ID = 38956]
2. the order of the autoregressive model [Option ID = 38955]
3. the order of the moving-average model [Option ID = 38957]
4. None of these [Option ID = 38958]
Correct Answer :-
- the order of the autoregressive model [Option ID = 38955]
Let us consider a Markov chain with state space S = {0,1,2,3} and associated probability transition matrix P = ((0, 0.5, 0, 0.5), (0.5, 0, 0.5, 0), (0, 0, 0.5, 0.5), (0, 0, 0.5, 0.5)) For a Markov chain starting from state 0, determine the expected number of visits that the chain makes to state 0 before reaching a recurrent class. [Question ID = 17220]
1. 4/3 [Option ID = 38874]
2. 1/2 [Option ID = 38871]
3. 3/4 [Option ID = 38873]
4. 2/3 [Option ID = 38872]
Correct Answer :-
- 4/3 [Option ID = 38874]

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The joint distribution of the number of deaths in a life table is [Question ID = 17250]
1. Binomial [Option ID = 38991]
2. Negative binomial [Option ID = 38993]
3. Multinomial [Option ID = 38992]
4. Normal [Option ID = 38994]
Correct Answer :-
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- Multinomial [Option ID = 38992]
Under Type I censoring
1. Number of failures is fixed
2. Number of failures is a random variable
3. Time to is fixed
4. Time to is a random variable
Which of the above is/are correct? [Question ID = 17218]
1. Both 1 and 3 [Option ID = 38865]
2. Both 1 and 4 [Option ID = 38866]
3. 4 only [Option ID = 38863]
4. Both 2 and 3 [Option ID = 38864]
Correct Answer :-
- Both 2 and 3 [Option ID = 38864]
If all the v - 1 non-zero characteristic roots of the C-matrix of a design are equal to ?, then each elementary contrast t1 - tj is estimated with a variance: [Question ID = 17226]
1. ?/s² [Option ID = 38895]
2. 2?/s² [Option ID = 38896]
3. s²/? [Option ID = 38898]
4. 2s²/? [Option ID = 38897]
Correct Answer :-
- 2?/s² [Option ID = 38896]

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It is decided to accept a product with variability ±3s. Assuming a normally distributed controlled process, the number of rejects, on an average, expected in a population of 1,00,000 is [Question ID = 17247]
1. 300 [Option ID-38982]
2. 200 [Option ID = 38979]
3. 250 [Option ID = 38980]
4. 270 [Option ID = 38981]
Correct Answer :-
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- 270 [Option ID = 38981]
Let X1, X2,...Xn be iid exp(?) random variables. Then the unbiased estimator of ? based on Y=Min(X1, X2,...Xn) is: [Question ID = 17254]
1. nY [Option ID = 39010]
2. Y/n [Option ID = 39008]
3. n/Y [Option ID = 39009]
4. Y [Option ID = 39007]
Correct Answer :-
- nY [Option ID = 39010]
The mean of a truncated standard
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This download link is referred from the post: DUET Last 10 Years 2011-2021 Question Papers With Answer Key || Delhi University Entrance Test conducted by the NTA

This post was last modified on 29 January 2020

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