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Model MCQs — UGC NET Statistics

500 degree-level questions · 50 per unit · Click "Show Answer" to reveal

Topics Covered

Probability and Distributions Real Analysis & Matrix Algebra Sampling Methods & Design of Experiments Estimation Theory Testing of Hypotheses Linear Estimation, Regression & Econometrics Time Series Multivariate Analysis Stochastic Processes

About these MCQs

500 multiple-choice questions calibrated to undergraduate / Master's degree level — the standard UGC NET expects. Each question has four options; click Show Answer for the correct option and (where useful) a one-line hint.

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1

Unit 1 — Probability and Distributions (50 MCQs)

Probability, RVs, distributions, MGF, joint distributions, CLT, LLN.

1. If $A$ and $B$ are independent events with $P(A)=0.3,\,P(B)=0.4$, then $P(A\cup B)$ equals:
  1. 0.58
  2. 0.70
  3. 0.42
  4. 0.12
Show Answer

A. $P(A\cup B)=0.3+0.4-0.12=0.58$.

2. Two cards are drawn without replacement from a standard deck. The probability that both are kings is:
  1. $1/169$
  2. $1/221$
  3. $1/52$
  4. $4/52$
Show Answer

B. $\frac{4}{52}\cdot\frac{3}{51}=\frac{1}{221}$.

3. Bayes' theorem expresses the posterior probability in terms of:
  1. Likelihood only
  2. Prior only
  3. Prior × Likelihood / Marginal
  4. Conditional / Joint
Show Answer

C. Posterior $=$ (prior $\times$ likelihood) / marginal: $P(B_k\mid A)=\dfrac{P(A\mid B_k)P(B_k)}{\sum_i P(A\mid B_i)P(B_i)}$.

4. The MGF of a Poisson distribution with mean $\lambda$ is:
  1. $e^{\lambda(e^t-1)}$
  2. $e^{\lambda t}$
  3. $(1-\lambda t)^{-1}$
  4. $e^{-\lambda+\lambda t}$
Show Answer

A.

5. If $X\sim N(0,1)$, then $E(X^4)$ equals:
  1. 1
  2. 2
  3. 3
  4. 6
Show Answer

C. Fourth central moment of $N(0,1)$ is $3\sigma^4=3$.

6. For $X\sim\text{Exp}(\lambda)$, the median is:
  1. $1/\lambda$
  2. $\ln 2/\lambda$
  3. $\lambda\ln 2$
  4. $2/\lambda$
Show Answer

B. Solve $1-e^{-\lambda m}=1/2$.

7. The Chebyshev inequality gives, for $k=2$:
  1. $P(|X-\mu|\ge 2\sigma)\le 1/4$
  2. $P(|X-\mu|\ge 2\sigma)\le 1/2$
  3. $P(|X-\mu|\ge 2\sigma)\ge 1/4$
  4. $P(|X-\mu|\ge 2\sigma)=1/4$
Show Answer

A.

8. If $X_1,\ldots,X_n$ are iid $N(\mu,\sigma^2)$, then $(n-1)S^2/\sigma^2$ follows:
  1. $\chi^2_n$
  2. $\chi^2_{n-1}$
  3. $F$-distribution
  4. $t_{n-1}$
Show Answer

B.

9. The characteristic function of $N(\mu,\sigma^2)$ is:
  1. $e^{\mu t+\sigma^2 t^2/2}$
  2. $e^{i\mu t-\sigma^2 t^2/2}$
  3. $e^{-i\mu t+\sigma^2 t^2/2}$
  4. $e^{i\mu t+\sigma^2 t^2/2}$
Show Answer

B.

10. If $X\sim\text{Bin}(n,p)$ and $np\to\lambda$ as $n\to\infty$, $p\to 0$, then $X$ tends in distribution to:
  1. $N(0,1)$
  2. $\text{Poisson}(\lambda)$
  3. $\text{Exp}(\lambda)$
  4. $\chi^2_1$
Show Answer

B.

11. The convergence in probability implies convergence in:
  1. $L^2$
  2. Distribution
  3. Almost sure
  4. Mean square
Show Answer

B. The chain of implications is a.s. $\Rightarrow$ in probability $\Rightarrow$ in distribution, so convergence in probability implies convergence in distribution (converse false in general).

12. The variance of the sample mean from $n$ iid observations with population variance $\sigma^2$ is:
  1. $\sigma^2$
  2. $\sigma^2/n$
  3. $n\sigma^2$
  4. $\sigma^2/\sqrt n$
Show Answer

B.

13. The pdf $f(x)=cx^2,\,0<x<3$ requires $c=$:
  1. $1/3$
  2. $1/9$
  3. $1/27$
  4. $2/9$
Show Answer

B. $\int_0^3 cx^2 dx=9c=1\Rightarrow c=1/9$.

14. If $X$ and $Y$ are independent, $\text{Var}(X+Y)$ equals:
  1. $\text{Var}(X)\text{Var}(Y)$
  2. $\text{Var}(X)+\text{Var}(Y)$
  3. $\text{Var}(X)-\text{Var}(Y)$
  4. $2\text{Var}(X)+\text{Var}(Y)$
Show Answer

B.

15. For a Geometric($p$) distribution, $E(X)$ equals (number of trials until first success):
  1. $p$
  2. $1/p$
  3. $1/(1-p)$
  4. $p/(1-p)$
Show Answer

B.

16. The CDF $F(x)$ of a continuous random variable is:
  1. Always continuous
  2. Always differentiable
  3. Right-continuous and non-decreasing
  4. Left-continuous
Show Answer

C.

17. For a Cauchy distribution:
  1. Mean exists, variance does not
  2. Neither mean nor variance exists
  3. Both exist
  4. Variance exists, mean does not
Show Answer

B. The Cauchy density has tails so heavy that $\int|x|f(x)\,dx$ diverges; hence neither the mean nor the variance exists.

18. If $X_1,X_2$ are iid $N(0,1)$, then $X_1^2+X_2^2$ has distribution:
  1. $\chi^2_2$
  2. $N(0,2)$
  3. $F_{1,1}$
  4. $t_2$
Show Answer

A.

19. Weak law of large numbers requires:
  1. iid and finite variance
  2. iid and finite mean (Khintchin)
  3. Identical distribution only
  4. Independence only
Show Answer

B.

20. CLT (Lindeberg–Lévy) requires:
  1. iid with finite variance
  2. iid Bernoulli only
  3. Independence with bounded variance
  4. iid with finite mean
Show Answer

A.

21. If $X\sim U(0,1)$ and $Y=-\ln X$, then $Y\sim$:
  1. $U(0,1)$
  2. $\text{Exp}(1)$
  3. $N(0,1)$
  4. $\text{Gamma}(2,1)$
Show Answer

B.

22. For Beta($\alpha,\beta$), the mean is:
  1. $\alpha\beta$
  2. $\alpha/(\alpha+\beta)$
  3. $\alpha+\beta$
  4. $1/\alpha\beta$
Show Answer

B.

23. The conditional density $f_{Y\mid X}(y\mid x)$ is defined when:
  1. $f_Y(y)>0$
  2. $f_X(x)>0$
  3. $f(x,y)=0$
  4. $X,Y$ are independent
Show Answer

B.

24. If $X,Y$ have joint pdf $f(x,y)=2$ on $0<x<y<1$, then $E(Y)$ equals:
  1. $1/3$
  2. $1/2$
  3. $2/3$
  4. $3/4$
Show Answer

C. $E(Y)=\int_0^1 y\cdot 2y\,dy=2/3$.

25. Markov's inequality states for non-negative $X$:
  1. $P(X\ge a)\le E(X)/a$
  2. $P(X\ge a)\ge E(X)/a$
  3. $P(X\ge a)=E(X)/a$
  4. $P(X\ge a)\le \text{Var}(X)/a$
Show Answer

A.

26. For $X\sim\text{Bin}(n,p)$, the skewness is zero when:
  1. $p=0$
  2. $p=1$
  3. $p=1/2$
  4. $p=1/4$
Show Answer

C. Binomial skewness is $(1-2p)/\sqrt{npq}$, which vanishes only at $p=1/2$, where the distribution is symmetric.

27. The MGF uniquely determines the distribution provided:
  1. $X$ is bounded
  2. It exists in a neighbourhood of 0
  3. $E(X)$ is finite
  4. $X$ is symmetric
Show Answer

B.

28. If $X_n\xrightarrow{d}c$ (constant), then $X_n\xrightarrow{P}c$:
  1. Always
  2. Only if $X_n$ is iid
  3. Never
  4. Only if variance is finite
Show Answer

A. Convergence in distribution to a constant is equivalent to convergence in probability to that constant (a special, stronger case).

29. The coefficient of variation of $N(\mu,\sigma^2)$ is:
  1. $\mu/\sigma$
  2. $\sigma/\mu$
  3. $\sigma^2/\mu$
  4. $\sqrt{\sigma}/\mu$
Show Answer

B.

30. If $X\sim\chi^2_n$, then $E(X)$ and $\text{Var}(X)$ are:
  1. $n,n$
  2. $n,2n$
  3. $2n,n$
  4. $n,n^2$
Show Answer

B. $\chi^2_n=\text{Gamma}(n/2,1/2)$, so $E(X)=n$ and $\text{Var}(X)=2n$.

31. The Slutsky theorem says if $X_n\xrightarrow{d}X,\,Y_n\xrightarrow{P}c$, then:
  1. $X_n Y_n\xrightarrow{P}cX$
  2. $X_n+Y_n\xrightarrow{d}X+c$
  3. $X_n/Y_n\xrightarrow{d}X$
  4. None
Show Answer

B.

32. The pdf of order statistic $X_{(k)}$ from $n$ iid Uniform(0,1) variables is:
  1. Beta$(k,n-k+1)$
  2. Beta$(k+1,n-k)$
  3. Gamma$(k,1)$
  4. $\chi^2_{2k}$
Show Answer

A. The $k$th order statistic of a Uniform$(0,1)$ sample is Beta$(k,n-k+1)$, with mean $k/(n+1)$.

33. For two independent Poisson variables with means $\lambda_1,\lambda_2$, $X_1+X_2$ has distribution:
  1. $\text{Poisson}(\lambda_1+\lambda_2)$
  2. $\text{Poisson}(\lambda_1\lambda_2)$
  3. $\text{Poisson}(\max)$
  4. Not Poisson
Show Answer

A.

34. The probability integral transform states: if $X$ has continuous CDF $F$, then $F(X)$ is:
  1. $U(0,1)$
  2. Standard normal
  3. Exponential
  4. Same as $X$
Show Answer

A.

35. If $X,Y$ are jointly normal with $\text{Cov}(X,Y)=0$, then:
  1. They are uncorrelated but not independent
  2. They are always independent
  3. $X+Y$ is not normal
  4. $X=Y$
Show Answer

B. For jointly normal, uncorrelated $\Rightarrow$ independent.

36. The De Moivre–Laplace approximation states $\text{Bin}(n,p)\approx$:
  1. $N(np,np)$
  2. $N(np,npq)$
  3. $N(p,pq/n)$
  4. $\text{Poisson}(np)$
Show Answer

B. De Moivre$-$Laplace: for large $n$, $\text{Bin}(n,p)\approx N(np,\,npq)$ (variance $npq$, not $np$).

37. The moment of $X^r$ exists iff $E|X|^r$ is:
  1. Zero
  2. Negative
  3. Finite
  4. Infinite
Show Answer

C.

38. The square of a Student's $t_n$ variate follows $F_{1,n}$; i.e. $t_n^2=F_{1,n}$. True or False?
  1. True
  2. False
  3. Only if $n=1$
  4. Only for large $n$
Show Answer

A. If $Z\sim N(0,1)$ and $V\sim\chi^2_n$ are independent, $t_n=Z/\sqrt{V/n}$, so $t_n^2=Z^2/(V/n)=\chi^2_1/(\chi^2_n/n)=F_{1,n}$.

39. The mode of $\text{Gamma}(\alpha,\beta)$ with $\alpha>1$ is:
  1. $\alpha/\beta$
  2. $(\alpha-1)/\beta$
  3. $\alpha\beta$
  4. $1/\beta$
Show Answer

B.

40. The conditional expectation $E(Y\mid X)$ is:
  1. A constant
  2. A function of $Y$
  3. A function of $X$
  4. Equal to $E(Y)$
Show Answer

C.

41. Law of iterated expectation: $E(Y)=$
  1. $E[E(Y\mid X)]$
  2. $\text{Var}(Y\mid X)$
  3. $E(X)E(Y)$
  4. $E(X\mid Y)$
Show Answer

A.

42. Variance decomposition: $\text{Var}(Y)=$
  1. $E[\text{Var}(Y\mid X)]+\text{Var}[E(Y\mid X)]$
  2. $\text{Var}[E(Y\mid X)]-E[\text{Var}(Y\mid X)]$
  3. $E[\text{Var}(Y\mid X)]$
  4. $\text{Var}[E(Y\mid X)]$
Show Answer

A.

43. For $X\sim\text{Bin}(10,0.5)$, the most probable value(s) are:
  1. 4 and 5
  2. 5 only
  3. 5 and 6
  4. 0 and 10
Show Answer

B. Mode = $\lfloor(n+1)p\rfloor=5$.

44. If $X\sim N(\mu,\sigma^2)$, the probability that $X$ lies within one SD of $\mu$ is approximately:
  1. 0.50
  2. 0.68
  3. 0.95
  4. 0.99
Show Answer

B.

45. Borel-Cantelli (first) Lemma: if $\sum P(A_n)<\infty$, then $P(A_n\text{ i.o.})=$
  1. 1
  2. 0
  3. 1/2
  4. Undefined
Show Answer

B.

46. For Negative Binomial$(r,p)$, $E(X)$ (no. of trials for $r$th success) is:
  1. $r$
  2. $r/p$
  3. $rp$
  4. $p/r$
Show Answer

B. For Negative Binomial (trials until the $r$th success), $E(X)=r/p$ and $\text{Var}(X)=rq/p^2$.

47. If $X,Y$ iid $\text{Exp}(\lambda)$, then $X+Y\sim$:
  1. $\text{Exp}(2\lambda)$
  2. $\text{Gamma}(2,\lambda)$
  3. $\chi^2_4$
  4. $N(2/\lambda,2/\lambda^2)$
Show Answer

B.

48. The continuity correction in approximating $\text{Bin}(n,p)$ by $N$ replaces $P(X\le k)$ by:
  1. $\Phi((k-np)/\sqrt{npq})$
  2. $\Phi((k+0.5-np)/\sqrt{npq})$
  3. $\Phi((k-0.5-np)/\sqrt{npq})$
  4. $\Phi(k)$
Show Answer

B.

49. For the multinomial distribution with parameters $(n,p_1,\ldots,p_k)$, $\text{Cov}(X_i,X_j)$ for $i\ne j$ is:
  1. $np_i p_j$
  2. $-np_i p_j$
  3. 0
  4. $p_i p_j$
Show Answer

B. Multinomial counts are negatively associated (they sum to $n$): $\text{Cov}(X_i,X_j)=-np_ip_j$ for $i\ne j$.

50. The standard normal distribution has kurtosis:
  1. 0
  2. 1
  3. 3
  4. $\pi$
Show Answer

C. (Excess kurtosis = 0; raw kurtosis = 3.)

2

Unit 2 — Real Analysis & Matrix Algebra (50 MCQs)

Sequences, series, continuity, differentiation, vector spaces, eigenvalues, quadratic forms.

1. The set $\mathbb Q$ of rationals is:
  1. Finite
  2. Countable
  3. Uncountable
  4. Empty
Show Answer

B.

2. The sequence $a_n=(1+1/n)^n$ converges to:
  1. 1
  2. $e$
  3. $2$
  4. $\infty$
Show Answer

B.

3. Cauchy criterion for convergence of sequence: $(x_n)$ converges iff:
  1. Bounded
  2. Monotonic
  3. For all $\varepsilon>0,\,\exists N$ s.t. $|x_m-x_n|<\varepsilon$ for $m,n\ge N$
  4. $x_n\to 0$
Show Answer

C.

4. The series $\sum 1/n^p$ converges iff:
  1. $p>0$
  2. $p\ge 1$
  3. $p>1$
  4. $p<1$
Show Answer

C.

5. Radius of convergence of $\sum x^n/n!$ is:
  1. 0
  2. 1
  3. $e$
  4. $\infty$
Show Answer

D.

6. Bolzano–Weierstrass theorem: every bounded sequence has:
  1. A limit
  2. A convergent subsequence
  3. A monotone subsequence only
  4. Cauchy subsequence only
Show Answer

B.

7. The function $f(x)=1/x$ on $(0,1)$ is:
  1. Uniformly continuous
  2. Continuous but not uniformly continuous
  3. Not continuous
  4. Bounded
Show Answer

B.

8. Mean Value Theorem (Lagrange) requires $f$ to be:
  1. Continuous on $[a,b]$, differentiable on $(a,b)$
  2. Differentiable everywhere
  3. Just continuous
  4. $f(a)=f(b)$
Show Answer

A.

9. Rolle's theorem requires additionally:
  1. $f$ monotone
  2. $f(a)=f(b)$
  3. $f$ bounded
  4. $f$ periodic
Show Answer

B.

10. L'Hôpital's rule applies to indeterminate forms:
  1. $0/0$ only
  2. $\infty/\infty$ only
  3. Both $0/0$ and $\infty/\infty$
  4. $0\cdot\infty$ only
Show Answer

C.

11. $\int_0^\infty e^{-x^2}dx$ equals:
  1. $\sqrt\pi$
  2. $\sqrt\pi/2$
  3. $\pi/2$
  4. 1
Show Answer

B.

12. The improper integral $\int_1^\infty 1/x^p\,dx$ converges iff:
  1. $p>0$
  2. $p>1$
  3. $p\ge 1$
  4. $p<1$
Show Answer

B.

13. The Taylor expansion of $\sin x$ around 0 starts as:
  1. $x+x^3/3!+\ldots$
  2. $x-x^3/3!+x^5/5!-\ldots$
  3. $1-x^2/2!+\ldots$
  4. $x^2/2+\ldots$
Show Answer

B.

14. For $f(x,y)=x^2+y^2$ at $(0,0)$: this is a:
  1. Local maximum
  2. Saddle point
  3. Local minimum
  4. Inflection
Show Answer

C.

15. Hessian determinant test at critical point: $H>0,\,f_{xx}<0$ implies:
  1. Local minimum
  2. Local maximum
  3. Saddle point
  4. Inconclusive
Show Answer

B.

16. Method of Lagrange multipliers solves:
  1. $\nabla f=0$
  2. $\nabla f=\lambda\nabla g$ and $g=0$
  3. $\nabla g=0$
  4. $f=g$
Show Answer

B.

17. A subset $W$ of vector space $V$ is a subspace iff:
  1. It contains $\mathbf 0$
  2. Closed under addition and scalar multiplication
  3. Has same dimension
  4. Is finite
Show Answer

B.

18. The dimension of $\mathbb R^3$ is:
  1. 1
  2. 2
  3. 3
  4. $\infty$
Show Answer

C.

19. Rank–nullity theorem: $\text{rank}(A)+\text{nullity}(A)=$
  1. Number of rows
  2. Number of columns
  3. $\det A$
  4. $\text{tr}(A)$
Show Answer

B.

20. For $n\times n$ matrix $A$, $\det A$ equals:
  1. Sum of eigenvalues
  2. Product of eigenvalues
  3. Trace
  4. Sum of diagonals only
Show Answer

B. The determinant equals the product of the eigenvalues: $\det A=\prod_i\lambda_i$.

21. $\text{tr}(A)$ equals:
  1. Sum of eigenvalues
  2. Product of eigenvalues
  3. $\det A$
  4. Rank
Show Answer

A. The trace equals the sum of the eigenvalues: $\text{tr}(A)=\sum_i\lambda_i$.

22. $\det(AB)$ equals:
  1. $\det A+\det B$
  2. $\det A\cdot\det B$
  3. $\det A-\det B$
  4. $|\det A-\det B|$
Show Answer

B.

23. A square matrix $A$ is invertible iff:
  1. $\det A=0$
  2. $\det A\ne 0$
  3. $\text{tr}(A)\ne 0$
  4. Symmetric
Show Answer

B.

24. Eigenvalues of a real symmetric matrix are:
  1. Always real
  2. Always complex
  3. Always positive
  4. Always zero
Show Answer

A. By the spectral theorem, a real symmetric matrix has all-real eigenvalues and an orthonormal eigenbasis.

25. Eigenvectors corresponding to distinct eigenvalues of a symmetric matrix are:
  1. Identical
  2. Orthogonal
  3. Parallel
  4. Linearly dependent
Show Answer

B.

26. Cayley–Hamilton theorem: a matrix satisfies:
  1. $A=A^T$
  2. Its own characteristic equation
  3. $AA^T=I$
  4. $A^2=A$
Show Answer

B.

27. Orthogonal matrix $A$ satisfies $\det A=$
  1. 0
  2. $\pm 1$
  3. Any value
  4. $+1$ only
Show Answer

B.

28. A positive definite matrix has:
  1. All eigenvalues 0
  2. All eigenvalues positive
  3. Some negative eigenvalues
  4. Determinant zero
Show Answer

B.

29. Sylvester's criterion: PD iff:
  1. All leading minors positive
  2. Trace positive
  3. Determinant positive
  4. All rows positive
Show Answer

A.

30. The system $Ax=b$ has a unique solution iff:
  1. $\text{rank}(A)=\text{rank}([A\mid b])=n$
  2. $\text{rank}(A)<n$
  3. $\det A=0$
  4. $b=0$
Show Answer

A.

31. Gram–Schmidt orthogonalization produces:
  1. Orthonormal basis from any spanning set
  2. Orthonormal basis from linearly independent set
  3. Eigenvalues
  4. Inverse
Show Answer

B.

32. For idempotent matrix ($A^2=A$), eigenvalues are:
  1. 0 or 1
  2. $\pm 1$
  3. 0 only
  4. Any real
Show Answer

A. $A^2=A$ forces $\lambda^2=\lambda$ on each eigenvalue, so every eigenvalue is $0$ or $1$.

33. For an idempotent matrix $A$, $\text{rank}(A)=$
  1. $\det A$
  2. $\text{tr}(A)$
  3. $n$
  4. 0
Show Answer

B. For an idempotent matrix, rank $=$ trace $=$ number of unit eigenvalues.

34. A nilpotent matrix has all eigenvalues:
  1. Equal to 1
  2. Equal to 0
  3. Positive
  4. Negative
Show Answer

B. $A^k=0$ forces $\lambda^k=0$, so every eigenvalue of a nilpotent matrix is $0$.

35. Cramer's rule applies only when:
  1. $\det A=0$
  2. $\det A\ne 0$
  3. $A$ symmetric
  4. $b=0$
Show Answer

B.

36. The set $[0,1]$ is:
  1. Countable
  2. Uncountable
  3. Finite
  4. Empty
Show Answer

B.

37. A monotone bounded sequence in $\mathbb R$:
  1. Diverges
  2. Converges
  3. Oscillates
  4. Has no limit
Show Answer

B.

38. Ratio test gives convergence when $\lim|a_{n+1}/a_n|$ is:
  1. $>1$
  2. $=1$
  3. $<1$
  4. 0 only
Show Answer

C.

39. The alternating series $\sum (-1)^n/n$:
  1. Converges absolutely
  2. Converges conditionally
  3. Diverges
  4. Equals 1
Show Answer

B. $\sum(-1)^n/n$ converges (Leibniz) but $\sum 1/n$ diverges, so it is conditionally, not absolutely, convergent.

40. Differentiability of $f$ at $a$ implies:
  1. Continuity at $a$
  2. Continuity nowhere
  3. $f(a)=0$
  4. $f'(a)=0$
Show Answer

A.

41. For a continuous function on $[a,b]$:
  1. Always differentiable
  2. Attains max and min
  3. Always monotone
  4. Always integrable as Lebesgue but not Riemann
Show Answer

B.

42. Span of $\{(1,0),(0,1),(1,1)\}$ in $\mathbb R^2$ is:
  1. $\mathbb R$
  2. $\mathbb R^2$
  3. A line
  4. $\{(0,0)\}$
Show Answer

B.

43. Rank of $\begin{pmatrix}1&2\\2&4\end{pmatrix}$ is:
  1. 0
  2. 1
  3. 2
  4. Undefined
Show Answer

B. The second row is twice the first, so the rows are dependent and the rank is $1$.

44. The eigenvalues of $\begin{pmatrix}2&0\\0&3\end{pmatrix}$ are:
  1. 2,3
  2. 0,1
  3. $\pm\sqrt 6$
  4. 5
Show Answer

A.

45. Two matrices are similar iff they share:
  1. Determinants only
  2. Trace, determinant, eigenvalues, char polynomial
  3. Just rank
  4. Just size
Show Answer

B. Similar matrices $B=P^{-1}AP$ have the same characteristic polynomial, so the same eigenvalues, trace and determinant; rank alone or size alone is far too little. Strictly these shared quantities are necessary, not sufficient: $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ and $I_2$ share all of them yet are not similar. Full similarity needs the same Jordan form, so B is the best of the four, and it is exactly right for diagonalizable matrices.

46. Quadratic form $x^TAx$ is negative definite iff:
  1. All eigenvalues positive
  2. All eigenvalues negative
  3. Mixed
  4. Zero
Show Answer

B.

47. The Schwarz inequality states:
  1. $|\langle x,y\rangle|\le \|x\|\|y\|$
  2. $|\langle x,y\rangle|\ge \|x\|\|y\|$
  3. $\langle x,y\rangle=\|x\|\|y\|$
  4. $\langle x,y\rangle=0$
Show Answer

A.

48. A function $f:\mathbb R\to\mathbb R$ is uniformly continuous on $[a,b]$ (compact) if:
  1. $f$ bounded
  2. $f$ continuous
  3. $f$ differentiable
  4. $f$ monotone
Show Answer

B. Heine–Cantor.

49. The double integral $\iint_R f(x,y)dA$ where $R$ is bounded can be evaluated by:
  1. Iterated integration
  2. Differentiation only
  3. Cramer's rule
  4. Always polar coordinates
Show Answer

A.

50. The characteristic polynomial of $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is:
  1. $\lambda^2-(a+d)\lambda+(ad-bc)$
  2. $\lambda^2+(a+d)\lambda-(ad-bc)$
  3. $\lambda^2-(ad-bc)\lambda+(a+d)$
  4. $\lambda^2+(ad-bc)$
Show Answer

A. Characteristic polynomial $=\lambda^2-(\text{tr})\lambda+\det=\lambda^2-(a+d)\lambda+(ad-bc)$.

3

Unit 3 — Sampling Methods & Design of Experiments (50 MCQs)

SRS, stratified, cluster, PPS, ANOVA, CRD, RBD, LSD, BIBD.

1. In SRSWOR, the inclusion probability of each unit is:
  1. $1/N$
  2. $n/N$
  3. $1/n$
  4. $N/n$
Show Answer

B.

2. The variance of $\bar y$ under SRSWOR is:
  1. $\sigma^2/n$
  2. $(1-f)S^2/n$
  3. $S^2/N$
  4. $f S^2/n$
Show Answer

B.

3. Finite population correction (fpc) is:
  1. $1-f$
  2. $1+f$
  3. $N/n$
  4. $n/N$
Show Answer

A.

4. Neyman allocation in stratified sampling allocates $n_h$ proportional to:
  1. $N_h$
  2. $N_h S_h$
  3. $S_h^2$
  4. $N_h^2$
Show Answer

B. Neyman allocation sets $n_h\propto N_hS_h$: sample more where a stratum is larger or more variable.

5. Optimum allocation under cost constraint allocates $n_h$ proportional to:
  1. $N_h S_h/\sqrt{c_h}$
  2. $N_h S_h c_h$
  3. $N_h/c_h$
  4. $S_h$
Show Answer

A.

6. Stratified sampling is more efficient than SRS when:
  1. Strata are homogeneous within
  2. Strata are heterogeneous within
  3. Always
  4. Never
Show Answer

A. Stratification gains most when strata are internally homogeneous, since between-strata variation is removed from the error.

7. In systematic sampling with $N=20,n=4$, the sampling interval $k$ is:
  1. 4
  2. 5
  3. 20
  4. 10
Show Answer

B.

8. Systematic sampling is efficient when intraclass correlation $\rho_w$ is:
  1. Positive and large
  2. Negative
  3. Zero
  4. Equal to 1
Show Answer

B. Systematic sampling beats SRS when units within the same sample are heterogeneous, i.e. intraclass correlation $\rho_w<0$.

9. Ratio estimator $\hat{\bar Y}_R=(\bar y/\bar x)\bar X$ is more efficient than SRS when:
  1. $\rho_{xy}>\tfrac{1}{2}\,C_x/C_y$
  2. $\rho=0$
  3. $X$ and $Y$ are independent
  4. $\bar x=\bar y$
Show Answer

A.

10. The regression estimator is always at least as efficient as:
  1. Ratio estimator
  2. SRS mean only
  3. Both ratio and SRS
  4. Neither
Show Answer

C.

11. Cluster sampling is efficient when:
  1. Clusters are heterogeneous within
  2. Clusters are homogeneous within
  3. $\rho>0$
  4. Both A and C imply inefficient
Show Answer

A. Clusters should be internally heterogeneous ($\rho<0$) so each surveyed cluster mirrors the whole population.

12. Horvitz–Thompson estimator of total is:
  1. $\sum y_i/\pi_i$ ($i\in s$)
  2. $\sum y_i p_i$
  3. $\sum y_i \pi_i$
  4. $\bar y N$
Show Answer

A.

13. PPS sampling stands for:
  1. Pure probability sampling
  2. Probability Proportional to Size
  3. Population Proportional Sampling
  4. Periodic Probability Sampling
Show Answer

B.

14. Hansen–Hurwitz estimator (PPSWR) uses:
  1. $\frac{1}{n}\sum y_i/p_i$
  2. $\sum y_i$
  3. $\bar y/n$
  4. $\sum y_i p_i$
Show Answer

A.

15. Yates–Grundy variance estimator is non-negative iff:
  1. $\pi_{ij}\ge\pi_i\pi_j$
  2. $\pi_{ij}\le\pi_i\pi_j$
  3. $\pi_{ij}=0$
  4. Always
Show Answer

B.

16. Double sampling is also known as:
  1. Two-stage sampling
  2. Two-phase sampling
  3. Cluster sampling
  4. Bootstrap sampling
Show Answer

B.

17. Fisher's three principles of design are:
  1. Randomisation, Replication, Local Control
  2. Symmetry, Balance, Order
  3. Independence, Identification, Estimation
  4. Blocking, Sampling, Testing
Show Answer

A.

18. In CRD with $t$ treatments and $r_i$ replicates, the error df is:
  1. $\sum r_i - t$
  2. $\sum r_i - 1$
  3. $t-1$
  4. $t r-1$
Show Answer

A.

19. In RBD with $b$ blocks and $t$ treatments, error df is:
  1. $(b-1)(t-1)$
  2. $bt-1$
  3. $b-1$
  4. $t-1$
Show Answer

A.

20. In a $t\times t$ Latin square design, error df is:
  1. $(t-1)^2$
  2. $(t-1)(t-2)$
  3. $t-1$
  4. $t^2-1$
Show Answer

B. LSD error df $=(t-1)(t-2)$: from the total $t^2-1$, subtract $3(t-1)$ for rows, columns and treatments.

21. The number of effects estimated in a $2^3$ factorial (excl. intercept) is:
  1. 3
  2. 6
  3. 7
  4. 8
Show Answer

C. 3 main + 3 two-factor + 1 three-factor.

22. In a $2^2$ factorial with $r$ replicates, error df is:
  1. $4(r-1)$
  2. $4r-1$
  3. $3$
  4. $r-1$
Show Answer

A.

23. Confounding sacrifices information on:
  1. Main effects
  2. Highest-order interactions usually
  3. Replication
  4. Blocks
Show Answer

B.

24. BIBD parameters satisfy:
  1. $bk=vr$, $\lambda(v-1)=r(k-1)$
  2. $bk=v\lambda$
  3. $r=k$
  4. $b=v$ always
Show Answer

A. Every BIBD satisfies $bk=vr$ and $\lambda(v-1)=r(k-1)$.

25. Fisher's inequality for BIBD states:
  1. $b\le v$
  2. $b\ge v$
  3. $b=v$
  4. $b=r$
Show Answer

B. Fisher's inequality: any BIBD has $b\ge v$ (at least as many blocks as treatments).

26. Connected design means:
  1. Block contrasts are estimable
  2. All treatment elementary contrasts are estimable
  3. Treatments are equally replicated
  4. Blocks are equal-sized
Show Answer

B.

27. Orthogonal design means:
  1. Treatment contrasts orthogonal to block contrasts
  2. $\det X'X=0$
  3. All treatments confounded
  4. $b=v$
Show Answer

A.

28. Efficiency factor of a BIBD is:
  1. $\lambda v/(rk)$
  2. $rk/(\lambda v)$
  3. $b/v$
  4. $r/k$
Show Answer

A. The efficiency factor of a BIBD is $E=\lambda v/(rk)$, always $<1$ relative to a complete-block design.

29. The ANOVA F-test compares:
  1. Between to within mean squares
  2. Within to total
  3. Between to total
  4. Total to error
Show Answer

A.

30. Yates' missing plot technique in RBD estimates missing value as:
  1. $(bB+tT-G)/[(b-1)(t-1)]$
  2. Mean of remaining
  3. 0
  4. $(bT-G)/(t-1)$
Show Answer

A.

31. Two-way ANOVA without interaction: error df with $a$ rows and $b$ cols is:
  1. $ab-1$
  2. $(a-1)(b-1)$
  3. $ab-a-b+1$
  4. Both B and C
Show Answer

D. Same.

32. For valid F-test in ANOVA, key assumption is:
  1. Errors are iid $N(0,\sigma^2)$
  2. Errors are exponential
  3. Equal sample sizes only
  4. $\sigma^2$ known
Show Answer

A.

33. Latin square design controls how many sources of variation?
  1. 1
  2. 2
  3. 3
  4. 0
Show Answer

B.

34. Number of plots in a $4\times 4$ LSD is:
  1. 4
  2. 8
  3. 16
  4. 32
Show Answer

C.

35. Total number of effect contrasts (incl. main + interactions) in a $2^n$ factorial:
  1. $2^n$
  2. $2^n-1$
  3. $n$
  4. $2n$
Show Answer

B.

36. In partial confounding with $r=2$ replicates, fraction of info lost on confounded effect is:
  1. 1
  2. 1/2
  3. 1/4
  4. 0
Show Answer

B.

37. Stratification works best on a variable that is:
  1. Independent of $Y$
  2. Highly correlated with $Y$
  3. Random
  4. Unmeasured
Show Answer

B.

38. The proportional allocation gives $n_h=$
  1. $nW_h$
  2. $nS_h$
  3. $nW_h S_h$
  4. $n/L$
Show Answer

A.

39. Murthy's estimator (PPSWOR) is unbiased and:
  1. Always has smaller variance than Hansen-Hurwitz
  2. Unordered
  3. Both A and B
  4. Only for $n=1$
Show Answer

B. Murthy's estimator is the unordered form of Des Raj's ordered estimator: averaging over the orders in which the same units could have been drawn (Rao–Blackwellisation) keeps it unbiased and makes it never worse than Des Raj's. That it always beats the Hansen–Hurwitz estimator is not a general result, so A, and with it C, cannot be taken as true.

40. Intra-block analysis recovers info only on:
  1. Block effects
  2. Treatment contrasts orthogonal to blocks
  3. All treatment contrasts (even confounded ones)
  4. Error
Show Answer

B.

41. Inter-block analysis uses information from:
  1. Within-block variation
  2. Block totals
  3. Treatment means
  4. Error only
Show Answer

B.

42. Recovery of inter-block information is useful in:
  1. Orthogonal designs
  2. BIBD (non-orthogonal)
  3. RBD
  4. CRD
Show Answer

B.

43. In a $4\times 4$ LSD, number of treatments equals:
  1. 4
  2. 8
  3. 16
  4. 12
Show Answer

A.

44. The Bose–Shrikhande–Parker result disproved Euler's conjecture on:
  1. BIBD
  2. Mutually orthogonal Latin squares
  3. Hadamard matrices
  4. PBIBD
Show Answer

B.

45. SRSWR vs SRSWOR — which is more efficient?
  1. WR
  2. WOR
  3. Same
  4. Depends on $N$
Show Answer

B.

46. Cluster size $M$ — variance increases as $\rho$ approaches:
  1. $-1$
  2. 0
  3. $+1$
  4. $+\infty$
Show Answer

C.

47. The Cochran condition for valid chi-square in contingency tables requires expected counts:
  1. $\ge 1$
  2. $\ge 5$ (most cells)
  3. $=0$
  4. $\le 5$
Show Answer

B.

48. Resolution III design confounds main effects with:
  1. 2-factor interactions
  2. 3-factor interactions
  3. Nothing
  4. Blocks
Show Answer

A.

49. In one-way ANOVA, total df is:
  1. $N-1$
  2. $N-k$
  3. $k-1$
  4. $k$
Show Answer

A.

50. The number of mutually orthogonal Latin squares of order $n$ is at most:
  1. $n$
  2. $n-1$
  3. $n+1$
  4. $n^2$
Show Answer

B. There are at most $n-1$ mutually orthogonal Latin squares of order $n$ (the maximum is reached when $n$ is a prime power).

4

Unit 4 — Estimation Theory (50 MCQs)

Unbiasedness, MLE, UMVUE, Cramér–Rao, sufficiency, Rao–Blackwell, Lehmann–Scheffé, intervals.

1. An estimator $T$ is unbiased for $g(\theta)$ if:
  1. $E(T)=g(\theta)\,\forall\theta$
  2. $\text{Var}(T)=0$
  3. $T$ is sufficient
  4. $T$ is consistent
Show Answer

A.

2. MSE of an estimator equals:
  1. Variance
  2. Bias
  3. Variance + Bias²
  4. Variance - Bias²
Show Answer

C.

3. Sample variance with denominator $n-1$ is unbiased for:
  1. $\sigma$
  2. $\sigma^2$
  3. $E(X^2)$
  4. $\mu$
Show Answer

B.

4. MLE is generally:
  1. Always unbiased
  2. Asymptotically efficient
  3. Has minimum variance
  4. Independent of sample
Show Answer

B. Under regularity conditions the MLE is consistent and asymptotically efficient, attaining the Cram\'er$-$Rao bound as $n\to\infty$.

5. Invariance property of MLE: if $\hat\theta$ is MLE of $\theta$, then MLE of $g(\theta)$ is:
  1. $g(\theta)$
  2. $g(\hat\theta)$
  3. $\hat\theta$
  4. Generally not exists
Show Answer

B.

6. The Method of Moments estimates parameters by equating:
  1. Sample and population moments
  2. Sample and population means only
  3. Variances only
  4. MLE and OLS
Show Answer

A.

7. Cramér–Rao lower bound for variance of unbiased estimator of $\theta$:
  1. $1/(n I(\theta))$
  2. $n I(\theta)$
  3. $I(\theta)/n$
  4. $\sigma^2/n$
Show Answer

A. For an unbiased estimator of $\theta$, the Cram\'er$-$Rao lower bound is $1/(nI(\theta))$.

8. Fisher information $I(\theta)=E[(\partial\log f/\partial\theta)^2]$ equals:
  1. $-E[\partial^2\log f/\partial\theta^2]$
  2. $E[\partial^2\log f/\partial\theta^2]$
  3. $\sigma^2$
  4. $1/\sigma^2$
Show Answer

A. (Under regularity).

9. Sufficiency: $T$ is sufficient for $\theta$ if:
  1. $E(T)=\theta$
  2. Conditional distribution of $X$ given $T$ is free of $\theta$
  3. Variance is min
  4. MLE
Show Answer

B.

10. Neyman–Fisher factorization: $T$ sufficient iff $f(x;\theta)=$
  1. $g(T(x),\theta)h(x)$
  2. $g(\theta)h(x)$
  3. $g(T(x))$
  4. $h(x)$
Show Answer

A.

11. For $X_i\sim N(\mu,\sigma^2)$, sufficient statistic for $(\mu,\sigma^2)$ is:
  1. $\bar X$ only
  2. $S^2$ only
  3. $(\bar X,S^2)$ or $(\sum X_i,\sum X_i^2)$
  4. $X_{(1)}$
Show Answer

C.

12. Rao–Blackwell theorem: conditioning unbiased $W$ on sufficient $T$ yields estimator with:
  1. Larger variance
  2. Smaller (or equal) variance
  3. Bias
  4. Same MSE
Show Answer

B. Rao$-$Blackwellization never increases variance: $\text{Var}(E(W\mid T))\le\text{Var}(W)$, with equality only if $W$ is already a function of $T$.

13. Lehmann–Scheffé theorem: if $T$ complete sufficient and $h(T)$ unbiased, then $h(T)$ is:
  1. Inadmissible
  2. UMVUE
  3. Inefficient
  4. Biased
Show Answer

B.

14. Basu's theorem: complete sufficient stat and ancillary stat are:
  1. Correlated
  2. Independent
  3. Equal
  4. Identical
Show Answer

B.

15. An ancillary statistic has distribution that:
  1. Depends on $\theta$
  2. Does not depend on $\theta$
  3. Is uniform
  4. Is normal
Show Answer

B.

16. For $U(0,\theta)$, complete sufficient statistic is:
  1. $\bar X$
  2. $X_{(n)}$
  3. $X_{(1)}$
  4. $\sum X_i$
Show Answer

B. For $U(0,\theta)$ the maximum $X_{(n)}$ is complete sufficient.

17. UMVUE of $\theta$ for $U(0,\theta)$ is:
  1. $X_{(n)}$
  2. $(n+1)X_{(n)}/n$
  3. $2\bar X$
  4. $X_{(1)}+X_{(n)}$
Show Answer

B. Since $E(X_{(n)})=\tfrac{n}{n+1}\theta$, the bias-correcting UMVUE is $\tfrac{n+1}{n}X_{(n)}$.

18. Confidence interval based on pivot $T$ requires $T$'s distribution to be:
  1. Free of all parameters
  2. Normal
  3. $t$-distributed
  4. Known
Show Answer

A.

19. 95% CI for $\mu$ with $\sigma$ unknown, $n$ small is based on:
  1. Standard normal
  2. $t_{n-1}$
  3. Chi-square
  4. $F$
Show Answer

B.

20. 100(1-$\alpha$)% CI for $\sigma^2$ is:
  1. $\left(\frac{(n-1)S^2}{\chi^2_{n-1,\alpha/2}},\frac{(n-1)S^2}{\chi^2_{n-1,1-\alpha/2}}\right)$
  2. $\bar X\pm t S/\sqrt n$
  3. $\pm z\sigma/\sqrt n$
  4. $S^2/n$
Show Answer

A.

21. For proportion $p$, large-sample 95% CI is:
  1. $\hat p\pm 1.96\sqrt{\hat p(1-\hat p)/n}$
  2. $\hat p\pm 1.96$
  3. $\hat p\pm \hat p/n$
  4. $\hat p\pm 1.96/n$
Show Answer

A.

22. UMVUE of $e^{-\lambda}$ in $X_i\sim$ Poisson is:
  1. $(1-1/n)^T$ where $T=\sum X_i$
  2. $e^{-\bar X}$
  3. $\bar X$
  4. $1/n$
Show Answer

A.

23. The minimal sufficient statistic is unique:
  1. Always
  2. Up to a one-to-one function
  3. Never
  4. Only for normal
Show Answer

B.

24. An estimator is consistent if:
  1. $T_n\xrightarrow{P}\theta$
  2. $T_n\xrightarrow{d}\theta$
  3. $E(T_n)=\theta$
  4. $\text{Var}(T_n)=0$
Show Answer

A. Consistency means the estimator converges in probability to the parameter: $T_n\xrightarrow{P}\theta$.

25. Sufficient condition for consistency:
  1. $E(T_n)\to\theta,\,\text{Var}(T_n)\to 0$
  2. Unbiasedness only
  3. Independence
  4. Normality
Show Answer

A.

26. Efficiency of unbiased estimator: $e(T)=$
  1. $\text{CRLB}/\text{Var}(T)$
  2. $\text{Var}(T)/\text{CRLB}$
  3. $\text{Var}(T)$
  4. $\text{Bias}^2$
Show Answer

A.

27. Distribution of $X_{(k)}$ from continuous $F$ has pdf involving:
  1. $F^{k-1}(1-F)^{n-k}f(x)$
  2. $F^k(1-F)^k$
  3. Just $f(x)$
  4. $F^n$
Show Answer

A.

28. EDF $F_n(x)=$
  1. $\frac{1}{n}\sum I(X_i\le x)$
  2. $\bar X$
  3. $F(x)$
  4. $\frac{1}{n}\sum X_i$
Show Answer

A.

29. Glivenko–Cantelli theorem: $\sup_x|F_n(x)-F(x)|$ converges to 0:
  1. In probability only
  2. Almost surely
  3. In distribution
  4. In mean square only
Show Answer

B. Glivenko$-$Cantelli gives uniform almost-sure convergence: $\sup_x|F_n(x)-F(x)|\xrightarrow{a.s.}0$.

30. Spearman's $\rho_s$ uses:
  1. Raw data
  2. Ranks
  3. Squared differences
  4. Both A and C
Show Answer

B.

31. Kendall's $\tau=$
  1. $(C-D)/\binom{n}{2}$
  2. $(C+D)/n$
  3. $C-D$
  4. $CD/n$
Show Answer

A.

32. Both rank correlations are invariant under:
  1. Linear transformations only
  2. Any monotone transformation
  3. Sample size
  4. None
Show Answer

B.

33. Spearman's formula with no ties: $\rho_s=$
  1. $1-\frac{6\sum d_i^2}{n(n^2-1)}$
  2. $\frac{6\sum d_i^2}{n(n^2-1)}$
  3. $1-\frac{\sum d_i}{n}$
  4. $\frac{\sum d_i^2}{n}$
Show Answer

A.

34. Method of moments estimator may not be:
  1. Unbiased
  2. Always exist
  3. Both A and B
  4. Linear
Show Answer

C.

35. MLE of $\sigma^2$ in $N(\mu,\sigma^2)$ is:
  1. $\frac{1}{n}\sum(X_i-\bar X)^2$
  2. $\frac{1}{n-1}\sum(X_i-\bar X)^2$
  3. $\bar X$
  4. $\sum X_i^2/n$
Show Answer

A.

36. UMVUE need not always exist; but if exists is:
  1. Unique
  2. Multiple
  3. Biased
  4. None
Show Answer

A.

37. For Bernoulli$(p)$, CRLB for unbiased estimator of $p$ is:
  1. $p(1-p)/n$
  2. $pq$
  3. $1/n$
  4. $p/n$
Show Answer

A.

38. For Normal $N(\mu,1)$, MLE of $\mu$:
  1. Is $\bar X$ and attains CRLB
  2. Is biased
  3. Inconsistent
  4. Doesn't exist
Show Answer

A.

39. The asymptotic distribution of MLE $\hat\theta$ is:
  1. $N(\theta,1/(nI(\theta)))$
  2. $\chi^2$
  3. $U(0,1)$
  4. Cauchy
Show Answer

A.

40. For two-sample CI for $\mu_1-\mu_2$ with equal variances, the pivot uses:
  1. Pooled $S_p^2$
  2. $\sigma_1^2,\sigma_2^2$ known
  3. Wilcoxon
  4. Sign test
Show Answer

A.

41. Confidence interval interpretation:
  1. $P(\theta\in\text{CI})=1-\alpha$ (for fixed $\theta$)
  2. $(1-\alpha)$ of such intervals contain $\theta$ in repeated sampling
  3. $\theta$ is random
  4. None
Show Answer

B.

42. The pivot used for variance interval is:
  1. $(n-1)S^2/\sigma^2\sim\chi^2_{n-1}$
  2. $(\bar X-\mu)/(S/\sqrt n)$
  3. $S^2/\sigma^2$
  4. $\bar X$
Show Answer

A.

43. For ratio of variances $\sigma_1^2/\sigma_2^2$, the pivot involves:
  1. $\chi^2$
  2. $F$
  3. $t$
  4. Normal
Show Answer

B.

44. UMVUE of $\mu$ in $N(\mu,\sigma^2)$ with both unknown is:
  1. $\bar X$
  2. $S^2$
  3. Median
  4. $X_{(1)}$
Show Answer

A.

45. For Bernoulli, $\sum X_i$ is:
  1. Sufficient and complete
  2. Ancillary
  3. Biased
  4. Inconsistent
Show Answer

A.

46. Asymptotic CI for $\theta$ uses:
  1. $\hat\theta\pm z_{\alpha/2}\sqrt{1/(nI(\hat\theta))}$
  2. $\hat\theta\pm t$
  3. Bootstrap only
  4. None
Show Answer

A.

47. Sample range $X_{(n)}-X_{(1)}$ in $U(\theta,\theta+1)$ is:
  1. Sufficient
  2. Ancillary
  3. UMVUE
  4. Biased for $\theta$
Show Answer

B.

48. The Delta Method gives asymptotic distribution of $g(\hat\theta)$:
  1. $N(g(\theta),[g'(\theta)]^2\text{Var}(\hat\theta))$
  2. $N(0,1)$
  3. $\chi^2$
  4. Cauchy
Show Answer

A.

49. Two independent unbiased estimators with variances $V_1,V_2$ — best linear combo has variance:
  1. $V_1+V_2$
  2. $V_1 V_2/(V_1+V_2)$
  3. $(V_1+V_2)/2$
  4. $\sqrt{V_1V_2}$
Show Answer

B. Inverse-variance weighting minimises variance: the best linear combination has variance $1/(1/V_1+1/V_2)=V_1V_2/(V_1+V_2)$.

50. For Poisson($\lambda$), MLE of $\lambda$ equals MoM equals:
  1. $\bar X$
  2. $S^2$
  3. $X_{(n)}$
  4. $\sum X_i^2/n$
Show Answer

A.

5

Unit 5 — Testing of Hypotheses (50 MCQs)

NP lemma, UMP/UMPU/UMPI, LRT, SPRT, chi-square, rank tests.

1. Type I error is the probability of:
  1. Rejecting $H_0$ when true
  2. Accepting $H_0$ when false
  3. Rejecting $H_1$ when true
  4. Accepting $H_1$ when false
Show Answer

A.

2. Power of a test equals:
  1. $\alpha$
  2. $\beta$
  3. $1-\beta$
  4. $1-\alpha$
Show Answer

C.

3. Neyman–Pearson lemma is for:
  1. Composite vs simple
  2. Simple vs simple
  3. Two-sided tests
  4. Sequential tests
Show Answer

B. The Neyman$-$Pearson lemma gives the most powerful test only for a simple null against a simple alternative.

4. NP-MP critical region is based on:
  1. $\bar X$
  2. Likelihood ratio $\ge k$
  3. Variance
  4. Sample range
Show Answer

B.

5. Monotone likelihood ratio in $T$ implies existence of UMP for:
  1. $H_0:\theta\le\theta_0$ vs $H_1:\theta>\theta_0$
  2. Two-sided
  3. Any test
  4. Only simple-simple
Show Answer

A. A monotone likelihood ratio in $T$ yields (Karlin$-$Rubin) a UMP one-sided test rejecting for large $T$.

6. Karlin–Rubin theorem requires:
  1. Symmetry
  2. MLR
  3. Sufficient stat
  4. Normal distribution
Show Answer

B.

7. For two-sided test, UMP usually:
  1. Exists
  2. Does not exist
  3. Equals LRT
  4. Is sign test
Show Answer

B. Two-sided tests generally admit no UMP test; the optimal choice within unbiased tests is the UMPU test.

8. UMPU stands for:
  1. Uniformly Most Powerful Unbiased
  2. Uniformly Most Powerful Universal
  3. Unbiased Most Powerful Universal
  4. None
Show Answer

A.

9. LRT statistic $\Lambda=$
  1. $\sup_{\Theta_0}L/\sup_\Theta L$
  2. $L(\hat\theta)$
  3. $\sup L/\inf L$
  4. $L/L_0$
Show Answer

A.

10. Wilks' theorem: $-2\log\Lambda\to$
  1. $N(0,1)$
  2. $\chi^2_r$
  3. $\chi^2_{n-1}$
  4. $F$
Show Answer

B. Wilks' theorem: $-2\log\Lambda\xrightarrow{d}\chi^2_r$ with $r=\dim\Theta-\dim\Theta_0$.

11. One-sample $t$-test for $H_0:\mu=\mu_0$ uses statistic:
  1. $(\bar X-\mu_0)/(S/\sqrt n)\sim t_{n-1}$
  2. $\bar X\sim N$
  3. $\chi^2$
  4. $F$
Show Answer

A.

12. Two-sample pooled $t$-test assumes:
  1. Equal variances
  2. Equal means
  3. Unequal sample sizes
  4. Independence only
Show Answer

A.

13. SPRT continues until:
  1. $\lambda_n\ge A$ or $\le B$
  2. $n\ge N$
  3. $\bar X>c$
  4. Always one observation
Show Answer

A.

14. SPRT bounds $A,B$ approximated by:
  1. $A=(1-\beta)/\alpha,B=\beta/(1-\alpha)$
  2. $A=\alpha/\beta$
  3. $A=1$
  4. $A=B$
Show Answer

A.

15. Wald's SPRT minimises:
  1. $\alpha,\beta$ jointly given expected sample size
  2. Sample size
  3. $E_{\theta_0}(N)+E_{\theta_1}(N)$ for given $\alpha,\beta$
  4. Variance
Show Answer

C.

16. Chi-square goodness-of-fit with $k$ cells and $r$ params estimated has df:
  1. $k$
  2. $k-1$
  3. $k-1-r$
  4. $k-r$
Show Answer

C. Goodness-of-fit df $=k-1-r$: one lost for the total constraint and $r$ for parameters estimated from the data.

17. Chi-square test of independence in $r\times c$ table has df:
  1. $rc-1$
  2. $(r-1)(c-1)$
  3. $rc$
  4. $r+c-2$
Show Answer

B. For an $r\times c$ table, the test of independence has $(r-1)(c-1)$ degrees of freedom.

18. Yates' continuity correction is applied for:
  1. $2\times 2$ tables
  2. Large samples
  3. Multinomial
  4. One-sample
Show Answer

A.

19. Sign test for median uses:
  1. $S^+\sim\text{Bin}(n,1/2)$
  2. Normal
  3. $t$
  4. $\chi^2$
Show Answer

A.

20. Wilcoxon signed rank test assumes:
  1. Symmetry of distribution
  2. Normality
  3. Independence only
  4. Equal variances
Show Answer

A.

21. Mann–Whitney U is equivalent to:
  1. Sign test
  2. Wilcoxon rank-sum
  3. Kruskal–Wallis
  4. Friedman
Show Answer

B.

22. Kruskal–Wallis is for $k$ groups; under $H_0$:
  1. $H\sim\chi^2_k$
  2. $H\sim\chi^2_{k-1}$
  3. $H\sim F$
  4. $H\sim N$
Show Answer

B.

23. For $k=2$, Kruskal–Wallis equals:
  1. Mann–Whitney $Z^2$
  2. $t^2$
  3. $F$
  4. Sign test
Show Answer

A.

24. Ansari–Bradley test is for:
  1. Location
  2. Scale
  3. Both
  4. Independence
Show Answer

B.

25. van der Waerden test uses scores:
  1. Wilcoxon ranks
  2. Normal scores
  3. Rank squares
  4. Median scores
Show Answer

B.

26. A test is unbiased if:
  1. Power $\ge\alpha$ throughout $H_1$
  2. Estimator is unbiased
  3. $E(T)=0$
  4. None
Show Answer

A.

27. The $p$-value is:
  1. $P(H_0)$
  2. $P(\text{data}\mid H_0)$ in extreme region
  3. $P(\text{reject})$
  4. $\alpha$
Show Answer

B.

28. Power increases with:
  1. Larger $n$
  2. Smaller $\sigma$
  3. Larger effect size
  4. All of the above
Show Answer

D.

29. For testing variance $\sigma^2$ in $N$, statistic $(n-1)S^2/\sigma_0^2$ follows:
  1. $\chi^2_{n-1}$
  2. $t$
  3. $F$
  4. Normal
Show Answer

A.

30. For equality of two variances, the test stat is:
  1. $S_1^2/S_2^2\sim F$
  2. $S_1^2-S_2^2$
  3. $\chi^2$
  4. $t$
Show Answer

A.

31. Likelihood ratio test for $H_0:\mu=\mu_0$ in $N(\mu,\sigma^2)$ both unknown gives:
  1. $z$-test
  2. $t$-test
  3. $\chi^2$ test
  4. Sign test
Show Answer

B.

32. Average sample number (ASN) in SPRT:
  1. Is fixed
  2. Depends on $\theta$
  3. Always small
  4. Equal to $n$
Show Answer

B.

33. Operating characteristic (OC) curve gives:
  1. $P(\text{reject}\mid\theta)$
  2. $P(\text{accept }H_0\mid\theta)$
  3. $\alpha$
  4. $\beta$
Show Answer

B.

34. Most powerful test among size-$\alpha$ tests for simple-simple is given by:
  1. $\chi^2$ test
  2. NP lemma
  3. Wald test
  4. Score test
Show Answer

B.

35. Wald test statistic:
  1. $(\hat\theta-\theta_0)^2/\text{Var}(\hat\theta)$
  2. $-2\log\Lambda$
  3. Score function
  4. $\bar X$
Show Answer

A.

36. Score test (Rao) uses:
  1. $U(\theta_0)^2/I(\theta_0)$
  2. $\hat\theta$
  3. $\Lambda$
  4. $\bar X$
Show Answer

A.

37. Wald, LRT, Score test are:
  1. Identical in small samples
  2. Asymptotically equivalent under $H_0$
  3. Always different
  4. Equal to $t$-test
Show Answer

B. The Wald, likelihood-ratio and Rao score tests are asymptotically equivalent under $H_0$ (and under local alternatives).

38. For chi-square independence in $2\times 2$ table, df is:
  1. 1
  2. 2
  3. 3
  4. 4
Show Answer

A.

39. Number of comparisons in pairwise $t$-tests across $k$ groups:
  1. $k$
  2. $\binom{k}{2}$
  3. $k-1$
  4. $2k$
Show Answer

B.

40. Bonferroni correction divides $\alpha$ by:
  1. $n$
  2. $k$
  3. Number of comparisons
  4. $\sqrt n$
Show Answer

C.

41. Median test for $k$ samples:
  1. Chi-square
  2. Sign-based
  3. Mood
  4. Wilcoxon
Show Answer

A. The $k$-sample median test finds the combined median, counts in each sample the observations above and below it, and tests the resulting $2\times k$ table with a $\chi^2$ statistic on $k-1$ degrees of freedom. (It is sometimes called Mood's median test, but "Mood's test" on its own usually means his test for scale, so the defining answer is the chi-square.)

42. Linear rank test $T=\sum c_i a(R_i)$ with Wilcoxon scores $a(i)=i$ targets:
  1. Scale shift
  2. Location shift
  3. Independence
  4. Goodness of fit
Show Answer

B.

43. Sign test ignores observations equal to median; treatment is:
  1. Include them as +
  2. Drop them
  3. Include as $-$
  4. Add 0.5
Show Answer

B.

44. For Mann–Whitney, under $H_0,\,E(U)=$
  1. $mn/2$
  2. $mn$
  3. $m+n$
  4. $mn(m+n+1)/12$
Show Answer

A. Under $H_0$ (identical distributions), the Mann$-$Whitney statistic has mean $E(U)=mn/2$.

45. Var$(U)$ in Mann–Whitney:
  1. $mn(m+n+1)/12$
  2. $mn/2$
  3. $mn$
  4. $(m+n)/12$
Show Answer

A.

46. Asymptotic distribution under $H_0$ of $\chi^2$ goodness-of-fit is:
  1. $\chi^2$ with appropriate df
  2. $F$
  3. Normal
  4. $t$
Show Answer

A.

47. A composite null hypothesis is:
  1. $\theta=\theta_0$
  2. $\theta\ne\theta_0$
  3. $\theta\le\theta_0$ (or interval)
  4. $\theta=0$
Show Answer

C.

48. Power function $\beta(\theta)$ for size-$\alpha$ test satisfies:
  1. $\beta(\theta_0)\le\alpha$
  2. $\beta(\theta)$ always 1
  3. $\beta(\theta)=0$
  4. $\beta(\theta)\ge\alpha$
Show Answer

A.

49. Critical value increases as $\alpha$:
  1. Increases
  2. Decreases
  3. Stays
  4. Equal $\beta$
Show Answer

B. Smaller $\alpha\Rightarrow$ larger critical value.

50. For testing $H_0:p=0.5$ in Binomial$(n,p)$, exact test uses:
  1. Binomial distribution
  2. Normal approximation
  3. Both A and B (depending on $n$)
  4. $t$ test
Show Answer

A. An exact test uses the binomial distribution itself: under $H_0$ the number of successes is Binomial$(n,0.5)$, and the p-value is a sum of its probabilities. The normal approximation is a large-sample convenience, not an exact test, so B and C do not describe it.

6

Unit 6 — Linear Estimation, Regression & Econometrics (50 MCQs)

OLS, GLS, Gauss–Markov, multicollinearity, heteroscedasticity, autocorrelation, IV, 2SLS.

1. Under Gauss–Markov assumptions, OLS is:
  1. BLUE
  2. MVUE always
  3. Biased
  4. Inconsistent
Show Answer

A. Under the Gauss$-$Markov assumptions OLS is BLUE $-$ minimum variance among all linear unbiased estimators.

2. The OLS estimator of $\boldsymbol\beta$ is:
  1. $(X'X)^{-1}X'y$
  2. $X'y$
  3. $X(X'X)^{-1}y$
  4. $y'X$
Show Answer

A.

3. Hat matrix $H=X(X'X)^{-1}X'$ has trace equal to:
  1. $n$
  2. $p$ (number of parameters)
  3. 1
  4. 0
Show Answer

B. $\text{tr}(H)=\text{tr}(X(X'X)^{-1}X')=\text{tr}(I_p)=p$, the number of estimated parameters.

4. Unbiased estimator of $\sigma^2$ in linear model:
  1. SSE/$n$
  2. SSE/$(n-p)$
  3. SSE/$p$
  4. SST/$n$
Show Answer

B.

5. $R^2$ equals:
  1. SSR/SST
  2. SSE/SST
  3. SSR/SSE
  4. $1-$SSR/SST
Show Answer

A.

6. Adjusted $R^2$ penalises for:
  1. $n$
  2. Number of predictors
  3. $\sigma$
  4. Skewness
Show Answer

B.

7. Adjusted $R^2$ formula:
  1. $1-(1-R^2)(n-1)/(n-p)$
  2. $R^2/n$
  3. $1-R^2/n$
  4. $(1-R^2)p$
Show Answer

A. $R^2_{adj}=1-(1-R^2)\dfrac{n-1}{n-p}$; unlike $R^2$ it can fall when a useless predictor is added.

8. The slope $\hat\beta_1$ in simple linear regression equals:
  1. $S_{xy}/S_{xx}$
  2. $\bar y/\bar x$
  3. $r\cdot s_y/s_x$ (correlation form)
  4. Both A and C
Show Answer

D.

9. Variance of $\hat\beta_1$ in SLR:
  1. $\sigma^2/S_{xx}$
  2. $\sigma^2/n$
  3. $\sigma^2$
  4. $1/n$
Show Answer

A.

10. $F$-test for overall regression has df:
  1. $(p,n-p)$
  2. $(p-1,n-p)$
  3. $(n-1,p)$
  4. $(n,p)$
Show Answer

B.

11. Multicollinearity inflates:
  1. Bias
  2. Standard errors
  3. $R^2$
  4. Mean
Show Answer

B.

12. VIF threshold of concern is typically:
  1. $\gt 1$
  2. $\gt 5$
  3. $\gt 10$
  4. $\gt 100$
Show Answer

C.

13. Heteroscedasticity makes OLS:
  1. Biased
  2. Inefficient but unbiased
  3. Inconsistent
  4. BLUE
Show Answer

B. Heteroscedasticity leaves OLS unbiased and consistent but no longer efficient, and the usual standard errors are biased.

14. Breusch–Pagan test for heteroscedasticity uses:
  1. $nR^2$ of $\hat e^2$ on $X$
  2. $F$ stat
  3. $\chi^2_{n-1}$
  4. Sign test
Show Answer

A.

15. White's test additionally includes:
  1. Squared and cross terms of $X$
  2. Lagged residuals
  3. $Y^2$
  4. None
Show Answer

A.

16. Autocorrelation in errors makes:
  1. OLS BLUE
  2. OLS unbiased but inefficient
  3. OLS biased
  4. $R^2=0$
Show Answer

B.

17. Durbin–Watson statistic close to 2 indicates:
  1. Positive autocorrelation
  2. Negative autocorrelation
  3. No autocorrelation
  4. Heteroscedasticity
Show Answer

C. $d\approx 2(1-\hat\rho)$, so $d\approx 2$ corresponds to $\hat\rho\approx 0$, i.e. no first-order autocorrelation.

18. Durbin–Watson $d\approx 2(1-\hat\rho)$ where $\hat\rho$ is:
  1. Sample correlation of residuals
  2. Mean residual
  3. $\sigma^2$
  4. $R^2$
Show Answer

A.

19. GLS estimator with cov $V$:
  1. $(X'V^{-1}X)^{-1}X'V^{-1}y$
  2. $(X'X)^{-1}X'y$
  3. $X'y$
  4. $V^{-1}y$
Show Answer

A.

20. Weighted least squares with weights $w_i\propto$
  1. $\sigma_i^2$
  2. $1/\sigma_i^2$
  3. $x_i$
  4. $1/x_i$
Show Answer

B.

21. Cochrane–Orcutt transforms variables to fix:
  1. Heteroscedasticity
  2. AR(1) autocorrelation
  3. Multicollinearity
  4. Bias
Show Answer

B.

22. To include a categorical variable with $k$ levels, you create:
  1. $k$ dummies
  2. $k-1$ dummies (with reference)
  3. $k+1$ dummies
  4. Just one variable
Show Answer

B.

23. Dummy variable trap occurs when:
  1. All $k$ dummies included with intercept
  2. $k-1$ dummies included
  3. No intercept
  4. $k=2$
Show Answer

A.

24. Logistic regression models:
  1. $E(Y\mid X)$ directly
  2. $\log[p/(1-p)]=x'\beta$
  3. $Y$ as normal
  4. $\log Y=x'\beta$
Show Answer

B.

25. Logistic regression coefficients are interpreted as:
  1. Marginal effects
  2. Log odds ratios
  3. Variances
  4. Probabilities
Show Answer

B.

26. Errors-in-variables model results in OLS:
  1. Unbiased
  2. Attenuation bias toward 0
  3. Inflated slope
  4. Zero variance
Show Answer

B. Classical measurement error in $X$ attenuates the slope toward zero by the factor $\sigma_X^2/(\sigma_X^2+\sigma_u^2)$.

27. Instrumental Variable estimator $\hat\beta_{IV}=$
  1. $(Z'X)^{-1}Z'y$
  2. $(X'X)^{-1}X'y$
  3. $(X'Z)^{-1}X'y$
  4. $Z'y$
Show Answer

A.

28. Valid instrument $Z$ must be:
  1. Correlated with $X$, uncorrelated with $\varepsilon$
  2. Correlated with $\varepsilon$
  3. Equal to $X$
  4. Random noise
Show Answer

A.

29. Two-Stage Least Squares involves:
  1. OLS twice on same data
  2. Regress endogenous on instruments, then OLS
  3. GLS
  4. WLS
Show Answer

B.

30. In simultaneous equations, equation is exactly identified iff:
  1. $K-k=m-1$
  2. $K-k>m-1$
  3. $K-k<m-1$
  4. $K=k$
Show Answer

A. The order condition for exact identification is $K-k=m-1$ (excluded exogenous variables just match included endogenous ones).

31. Over-identified equation has:
  1. $K-k\gt m-1$
  2. $K-k=m-1$
  3. $K-k\lt m-1$
  4. No instruments
Show Answer

A.

32. Rank condition for identification needs:
  1. Coefficient matrix of excluded variables to have rank $M-1$
  2. Determinant zero
  3. Trace one
  4. None
Show Answer

A.

33. $k$-class estimator with $k=0$ equals:
  1. OLS
  2. 2SLS
  3. LIML
  4. GLS
Show Answer

A.

34. $k$-class with $k=1$ equals:
  1. OLS
  2. 2SLS
  3. LIML
  4. Indirect LS
Show Answer

B.

35. LIML stands for:
  1. Linear Identification Method
  2. Limited Information Maximum Likelihood
  3. Local Iterated MLE
  4. Logistic Iterated ML
Show Answer

B.

36. Reduced form parameters are always:
  1. Estimable by OLS
  2. Endogenous
  3. Biased
  4. Confounded
Show Answer

A.

37. The Chow test detects:
  1. Structural break
  2. Heteroscedasticity
  3. Multicollinearity
  4. Normality
Show Answer

A.

38. Robust (White) standard errors correct for:
  1. Heteroscedasticity
  2. Autocorrelation
  3. Multicollinearity
  4. Outliers
Show Answer

A.

39. Newey–West SE corrects for:
  1. Heteroscedasticity only
  2. Both heteroscedasticity and autocorrelation
  3. Multicollinearity
  4. Heteroskedasticity in multivariate
Show Answer

B.

40. The mean function in classical linear regression is:
  1. $E(Y\mid X)=X\beta$
  2. $E(Y)=\bar X$
  3. $E(Y\mid X)=\sigma^2$
  4. None
Show Answer

A.

41. Residual sum of squares is:
  1. $\sum(\hat y_i-\bar y)^2$
  2. $\sum(y_i-\hat y_i)^2$
  3. $\sum y_i^2$
  4. $n\sigma^2$
Show Answer

B.

42. Stochastic regressor independent of error: OLS is:
  1. Biased
  2. Unbiased and consistent
  3. Inconsistent
  4. Constant
Show Answer

B.

43. Mixed estimator (Theil–Goldberger):
  1. $(X'X+R'\Phi^{-1}R)^{-1}(X'y+R'\Phi^{-1}r)$
  2. $(X'X)^{-1}X'y$
  3. $X'y$
  4. $R'r$
Show Answer

A.

44. Ridge regression adds penalty:
  1. $\lambda\sum|\beta_j|$
  2. $\lambda\sum\beta_j^2$
  3. $\lambda\sum\beta_j$
  4. None
Show Answer

B.

45. LASSO uses penalty:
  1. $L_1$
  2. $L_2$
  3. $L_0$
  4. Both A and B
Show Answer

A.

46. F-test for $H_0:R\beta=r$ uses:
  1. $(\text{SSE}_R-\text{SSE}_F)/q\,/(\text{SSE}_F/(n-p))$
  2. $t$ stat
  3. $\chi^2$
  4. $z$
Show Answer

A.

47. The intercept in regression equals (when both centred):
  1. $\bar y-\hat\beta\bar x$
  2. 0
  3. $\bar y$
  4. $\bar x$
Show Answer

A.

48. Total sum of squares (TSS) decomposes as:
  1. SSR + SSE
  2. SSR + Variance
  3. SSE + Bias
  4. SSR $\times$ SSE
Show Answer

A.

49. With $n=p$, $R^2$ equals:
  1. 0
  2. 1 (perfect fit)
  3. 0.5
  4. Undefined
Show Answer

B. With $n=p$ the fitted plane passes through every point, so SSE $=0$ and $R^2=1$ (perfect but meaningless fit).

50. Endogeneity arises if:
  1. $\text{Cov}(X,\varepsilon)\ne 0$
  2. $X$ correlated with $Y$
  3. Errors normal
  4. $E(\varepsilon)=0$
Show Answer

A.

7

Unit 7 — Time Series (50 MCQs)

ACF, PACF, stationarity, AR, MA, ARMA, ARIMA, spectral density.

1. A time series is strictly stationary if:
  1. Mean is constant
  2. Joint distribution invariant under time shift
  3. Variance constant
  4. Both A and C
Show Answer

B.

2. Weak (covariance) stationarity requires:
  1. Constant mean and autocovariance depending on lag only
  2. Normality
  3. Independence
  4. Linearity
Show Answer

A.

3. For Gaussian process, weak stationarity implies:
  1. Strict stationarity
  2. Independence
  3. Ergodicity
  4. None
Show Answer

A.

4. ACF of white noise:
  1. 1 at lag 0, 0 elsewhere
  2. 1 always
  3. 0 always
  4. Decays exponentially
Show Answer

A.

5. ACF of MA($q$) cuts off after lag:
  1. $q$
  2. $q-1$
  3. $q+1$
  4. Never
Show Answer

A. An MA($q$) has ACF that cuts off after lag $q$, while its PACF tails off $-$ the identifying signature of MA models.

6. PACF of AR($p$) cuts off after lag:
  1. $p-1$
  2. $p$
  3. $p+1$
  4. Never
Show Answer

B. An AR($p$) has PACF that cuts off after lag $p$, while its ACF tails off.

7. For AR(1) $\phi=0.5$, $\rho_k=$
  1. $0.5$
  2. $0.5^k$
  3. $1$
  4. $0$
Show Answer

B.

8. AR(1) is stationary iff:
  1. $|\phi|=1$
  2. $|\phi|<1$
  3. $\phi>0$
  4. $\phi<0$
Show Answer

B. AR(1) is stationary iff $|\phi|<1$, equivalently the root of $1-\phi B=0$ lies outside the unit circle.

9. MA process is always:
  1. Stationary
  2. Non-stationary
  3. Invertible
  4. Both A and C
Show Answer

A.

10. MA(1) is invertible iff:
  1. $|\theta|<1$
  2. $|\theta|>1$
  3. $\theta=0$
  4. Always
Show Answer

A.

11. ARMA($p,q$) stationary iff:
  1. Roots of MA polynomial outside unit circle
  2. Roots of AR polynomial outside unit circle
  3. Both A and B
  4. Neither
Show Answer

B. ARMA stationarity depends only on the AR polynomial roots lying outside the unit circle (MA part governs invertibility).

12. ARMA invertible iff:
  1. MA roots outside unit circle
  2. AR roots inside unit circle
  3. Mean is zero
  4. None
Show Answer

A.

13. Yule–Walker equations relate AR coefficients to:
  1. ACF values
  2. Eigenvalues
  3. White noise variance
  4. Spectral density
Show Answer

A.

14. Random walk $Y_t=Y_{t-1}+\varepsilon_t$ has:
  1. Stationary process
  2. Variance growing as $t$
  3. $E(Y_t)=0$ always
  4. Both B and C
Show Answer

B. Writing $Y_t=Y_0+\varepsilon_1+\dots+\varepsilon_t$ gives $E(Y_t)=Y_0$ and $\text{Var}(Y_t)=t\sigma^2$, which grows with $t$, so the process is not stationary. $E(Y_t)$ is $0$ only if the walk starts at $Y_0=0$, which the question does not say, so C, and with it D, is not right.

15. ARIMA$(p,d,q)$ becomes stationary after differencing $d$ times. $d$ for random walk is:
  1. 0
  2. 1
  3. 2
  4. $\infty$
Show Answer

B.

16. Dickey–Fuller test detects:
  1. Heteroscedasticity
  2. Unit root
  3. Structural break
  4. Seasonality
Show Answer

B.

17. Box–Jenkins approach proceeds in steps:
  1. Identify, estimate, diagnose, forecast
  2. Forecast, identify
  3. Estimate, forecast
  4. Just fit OLS
Show Answer

A.

18. Ljung–Box test checks:
  1. Residuals are white noise
  2. Stationarity
  3. Normality
  4. Trend
Show Answer

A.

19. AIC for order selection penalises:
  1. 2$k$ for number of parameters
  2. $\log L$
  3. $n$
  4. None
Show Answer

A.

20. BIC penalises:
  1. $k\log n$
  2. $2k$
  3. $k$
  4. $2\log L$
Show Answer

A.

21. Wold decomposition expresses stationary process as:
  1. Deterministic + linear MA($\infty$)
  2. AR only
  3. Polynomial
  4. Sinusoidal
Show Answer

A.

22. Spectral density is the Fourier transform of:
  1. ACVF
  2. ACF
  3. Mean
  4. Series itself
Show Answer

A.

23. Spectral density of white noise is:
  1. 0
  2. Constant ($\sigma^2/2\pi$)
  3. Sinusoidal
  4. Exponential
Show Answer

B.

24. Periodogram is:
  1. Consistent estimator of spectral density
  2. Inconsistent estimator
  3. Always zero
  4. Sufficient stat
Show Answer

B. The raw periodogram is asymptotically unbiased but inconsistent $-$ its variance does not shrink as $T$ grows, so it must be smoothed.

25. Smoothing periodogram with window reduces:
  1. Bias
  2. Variance
  3. Mean
  4. Resolution alone
Show Answer

B.

26. AR(1) spectral density peaks at:
  1. $\omega=0$ if $\phi>0$
  2. $\omega=\pi$ if $\phi>0$
  3. $\omega=\pi/2$
  4. Always at 0
Show Answer

A. AR(1) with $\phi>0$ concentrates spectral power at low frequencies, giving a spectral-density peak at $\omega=0$.

27. AR(1) with $\phi<0$ peaks at:
  1. $\omega=0$
  2. $\omega=\pi$
  3. $\omega=\pi/2$
  4. None
Show Answer

B.

28. Forecasting AR(1) $h$-step ahead: $\hat Y_t(h)=$
  1. $\phi^h Y_t$
  2. $Y_t$
  3. 0
  4. $\bar Y$
Show Answer

A.

29. Forecast variance grows with horizon and approaches:
  1. 0
  2. Series variance
  3. $\sigma^2$
  4. $\infty$ always
Show Answer

B. For stationary process.

30. MA(1) forecast $\hat Y_t(h)$ for $h\ge 2$:
  1. $\theta\varepsilon_t$
  2. $0$
  3. $Y_t$
  4. $\bar Y$
Show Answer

B.

31. Exponential smoothing is optimal for which ARIMA:
  1. ARIMA(0,1,1)
  2. ARIMA(1,0,0)
  3. ARIMA(1,1,1)
  4. ARIMA(0,0,0)
Show Answer

A. Simple exponential smoothing is the optimal (minimum-MSE) forecast for an ARIMA(0,1,1) process.

32. Seasonal ARIMA notation includes:
  1. $(p,d,q)\times(P,D,Q)_s$
  2. $(p,d,q)$
  3. $(P,D,Q)$
  4. $(p,q)$
Show Answer

A.

33. Ergodicity in mean means:
  1. $\bar Y_T\xrightarrow{P}\mu$
  2. $Y_t$ iid
  3. $E(Y_t)=0$
  4. Stationary
Show Answer

A.

34. ACVF of AR(1): $\gamma_0=\sigma^2/(1-\phi^2)$ requires:
  1. $|\phi|=1$
  2. $|\phi|<1$
  3. $|\phi|\ne 0$
  4. Always
Show Answer

B.

35. Bartlett's formula approximates Var$(r_k)$ for large samples for white noise as:
  1. $1/T$
  2. $1/n$
  3. $\sigma^2$
  4. $1$
Show Answer

A.

36. Confidence band on sample ACF for white noise (95%):
  1. $\pm 1.96/\sqrt T$
  2. $\pm 1/T$
  3. $\pm 1.96$
  4. $\pm\sigma$
Show Answer

A.

37. ARMA(1,1) ACF: after lag 1, ACF satisfies:
  1. $\rho_k=\phi\rho_{k-1}$ ($k\ge 2$)
  2. $\rho_k=0$
  3. $\rho_k=\theta\rho_{k-1}$
  4. $\rho_k$ constant
Show Answer

A.

38. Augmented DF test extends DF by including:
  1. Lagged differences of $Y$
  2. $X$ variables
  3. Squared terms
  4. Seasonal dummies
Show Answer

A.

39. Cointegration refers to:
  1. Stationary linear combination of non-stationary series
  2. Two stationary series
  3. Independent series
  4. $X=Y$
Show Answer

A.

40. The MA representation of AR(1) is:
  1. $Y_t=\sum_{j=0}^\infty\phi^j\varepsilon_{t-j}$
  2. $Y_t=\varepsilon_t$
  3. Finite sum
  4. $Y_t=\phi^t$
Show Answer

A.

41. AR representation of MA(1) (when invertible):
  1. Converges with $\sum(-\theta)^j Y_{t-j}=\varepsilon_t$
  2. Doesn't exist
  3. Finite
  4. $\varepsilon_t=Y_t$
Show Answer

A.

42. Trend can be removed by:
  1. Differencing
  2. Detrending (regression)
  3. Both A and B
  4. None
Show Answer

C.

43. Seasonal differencing of period 12 removes:
  1. Yearly seasonality (monthly data)
  2. Trend
  3. Quarter effect
  4. Outliers
Show Answer

A.

44. Backshift operator $B$: $BY_t=$
  1. $Y_{t-1}$
  2. $Y_{t+1}$
  3. $Y_t$
  4. $Y_t-Y_{t-1}$
Show Answer

A.

45. Difference operator $(1-B)Y_t=$
  1. $Y_t-Y_{t-1}$
  2. $Y_t+Y_{t-1}$
  3. $Y_t$
  4. $Y_{t-1}$
Show Answer

A.

46. AR(2): $Y_t=\phi_1Y_{t-1}+\phi_2 Y_{t-2}+\varepsilon_t$ stationary iff roots of $1-\phi_1z-\phi_2z^2=0$ lie:
  1. Inside unit circle
  2. Outside unit circle
  3. On unit circle
  4. At 0
Show Answer

B.

47. The Wiener filter is used for:
  1. Optimal linear prediction/filtering
  2. Outlier detection
  3. Spectral peak
  4. None
Show Answer

A.

48. Sample mean of stationary series with absolutely summable ACVF is:
  1. Ergodic
  2. Inefficient
  3. Biased
  4. None
Show Answer

A.

49. Spectral representation theorem: any stationary process can be written as:
  1. Stochastic integral over frequencies
  2. Polynomial
  3. Sum of harmonic functions
  4. Both A and C (depending on context)
Show Answer

D.

50. The simplest model for white noise residuals confirms:
  1. Adequate model fit
  2. Inadequate
  3. Non-stationary
  4. Heteroscedastic
Show Answer

A.

8

Unit 8 — Multivariate Analysis (50 MCQs)

MVN, Wishart, Hotelling's $T^2$, PCA, discriminant, canonical correlation.

1. The MGF of $N_p(\boldsymbol\mu,\Sigma)$ is:
  1. $\exp(\mathbf t'\boldsymbol\mu+\frac{1}{2}\mathbf t'\Sigma\mathbf t)$
  2. $\exp(\mathbf t'\Sigma\mathbf t)$
  3. $\exp(\mathbf t')$
  4. $\boldsymbol\mu$
Show Answer

A.

2. Marginal of MVN is:
  1. Normal
  2. Generally not normal
  3. Uniform
  4. Chi-square
Show Answer

A.

3. For jointly normal $(X,Y)$, $\text{Cov}(X,Y)=0$ implies:
  1. Uncorrelated but dependent
  2. Independence
  3. Linear dependence
  4. Cannot say
Show Answer

B. For jointly normal variables, zero covariance implies full independence (a property special to the normal).

4. Conditional distribution of $X_1\mid X_2=x_2$ for MVN is:
  1. Normal
  2. Uniform
  3. Cauchy
  4. None
Show Answer

A.

5. Conditional mean for MVN:
  1. $\mu_1+\Sigma_{12}\Sigma_{22}^{-1}(x_2-\mu_2)$
  2. $\mu_1$
  3. $\mu_2$
  4. 0
Show Answer

A.

6. $(\mathbf X-\mu)'\Sigma^{-1}(\mathbf X-\mu)\sim$
  1. $\chi^2_p$
  2. $\chi^2_n$
  3. $F$
  4. $N$
Show Answer

A. $(\mathbf X-\mu)'\Sigma^{-1}(\mathbf X-\mu)\sim\chi^2_p$; this quadratic form is the squared Mahalanobis distance.

7. Sample mean vector $\bar{\mathbf X}$ from iid $N_p(\mu,\Sigma)$:
  1. $N_p(\mu,\Sigma/n)$
  2. $N_p(\mu,\Sigma)$
  3. $N_p(0,\Sigma)$
  4. $\chi^2$
Show Answer

A.

8. Sample covariance $\mathbf S=\frac{1}{n-1}\sum(\mathbf X_i-\bar{\mathbf X})(\mathbf X_i-\bar{\mathbf X})'$:
  1. $E(\mathbf S)=\Sigma$
  2. $E(\mathbf S)=\Sigma/n$
  3. Biased
  4. None
Show Answer

A.

9. $(n-1)\mathbf S$ has distribution:
  1. $W_p(n-1,\Sigma)$
  2. $W_p(n,\Sigma)$
  3. $\chi^2_{n-1}$
  4. $F$
Show Answer

A. $(n-1)\mathbf S\sim W_p(n-1,\Sigma)$ $-$ the multivariate analogue of $(n-1)S^2/\sigma^2\sim\chi^2_{n-1}$.

10. Wishart distribution is:
  1. Univariate
  2. Multivariate generalization of $\chi^2$
  3. Discrete
  4. Discrete normal
Show Answer

B.

11. Hotelling's $T^2$ relates to $F$ as:
  1. $\frac{n-p}{p(n-1)}T^2\sim F_{p,n-p}$
  2. $T^2\sim F$
  3. $T\sim t$
  4. $T^2=\chi^2$
Show Answer

A. $\dfrac{n-p}{p(n-1)}T^2\sim F_{p,n-p}$, which is how $T^2$ is referred to critical values.

12. For $p=1$, Hotelling's $T^2$ reduces to:
  1. $t^2$
  2. $z^2$
  3. $F$
  4. $\chi^2$
Show Answer

A.

13. Mahalanobis distance: $D^2=$
  1. $(\mu_1-\mu_2)'\Sigma^{-1}(\mu_1-\mu_2)$
  2. $\|\mu_1-\mu_2\|^2$
  3. $\text{tr}(\Sigma)$
  4. $\det\Sigma$
Show Answer

A.

14. Discriminant analysis projects onto direction:
  1. $\Sigma^{-1}(\mu_1-\mu_2)$
  2. $\mu_1+\mu_2$
  3. $\Sigma\mu_1$
  4. $\Sigma^{-1}\mu_1$
Show Answer

A.

15. Misclassification probability for equal-prior LDA is:
  1. $\Phi(-D/2)$
  2. $1/2$
  3. $\Phi(D)$
  4. $\Phi(D^2)$
Show Answer

A. With equal priors the optimal error rate is $\Phi(-D/2)$, decreasing as the Mahalanobis separation $D$ grows.

16. PCA finds:
  1. Eigenvectors of covariance matrix
  2. Eigenvalues of correlation matrix only
  3. Largest singular value of $X$
  4. All of the above
Show Answer

A. The principal components are the eigenvectors of the covariance matrix (or of the correlation matrix, when the variables are standardised), and their eigenvalues are the variances they explain. B is false because of its "only", and C names just the first singular value, which gives only the first component, so "all of the above" fails.

17. First PC explains:
  1. Maximum variance
  2. Minimum variance
  3. 50% variance
  4. None
Show Answer

A.

18. PCs are:
  1. Correlated
  2. Uncorrelated (orthogonal)
  3. Identical
  4. Random
Show Answer

B.

19. Total variance equals:
  1. Sum of eigenvalues
  2. Product
  3. $\det\Sigma$
  4. $\text{tr}\Sigma$ — same as A
Show Answer

D. Both A and "trace of $\Sigma$".

20. PCA on correlation matrix instead of covariance is used when:
  1. Variables on different scales
  2. Always
  3. Equal variances
  4. Never
Show Answer

A.

21. Canonical correlation between two sets is:
  1. Max correlation between linear combos
  2. Sum of correlations
  3. Average correlation
  4. 0
Show Answer

A.

22. Number of canonical correlations: $k=$
  1. $\min(p,q)$
  2. $\max(p,q)$
  3. $p+q$
  4. $pq$
Show Answer

A. There are $\min(p,q)$ canonical correlations between the two variable sets.

23. CCA reduces to multiple correlation when:
  1. $q=1$
  2. $p=1$
  3. $p=q$
  4. Both A and B
Show Answer

D.

24. Simple correlation $\rho_{12}=$
  1. $\sigma_{12}/\sqrt{\sigma_{11}\sigma_{22}}$
  2. $\sigma_{12}^2$
  3. $\sigma_{12}/\sigma_{11}$
  4. $\sigma_{12}\sigma_{11}$
Show Answer

A.

25. Partial correlation $\rho_{12\cdot 3}$ measures association between $X_1,X_2$ after removing:
  1. $X_3$ linear effect
  2. Means
  3. All other variables
  4. Nothing
Show Answer

A.

26. Multiple correlation $R_{1\cdot 23}$ measures:
  1. Correlation of $X_1$ with $X_2$
  2. Max correlation of $X_1$ with linear combo of $X_2,X_3$
  3. Average
  4. Sum
Show Answer

B.

27. Test of $H_0:\rho=0$ uses statistic:
  1. $r\sqrt{n-2}/\sqrt{1-r^2}\sim t_{n-2}$
  2. $z=r\sqrt n$
  3. $F$
  4. $\chi^2$
Show Answer

A.

28. Fisher's $z$-transform:
  1. $\frac{1}{2}\log[(1+r)/(1-r)]$
  2. $r^2$
  3. $\arctan r$
  4. $\log r$
Show Answer

A.

29. Wilks' $\Lambda$ for MANOVA:
  1. $|\mathbf W|/|\mathbf B+\mathbf W|$
  2. $|\mathbf B|/|\mathbf W|$
  3. $\text{tr}(\mathbf B\mathbf W^{-1})$
  4. None
Show Answer

A.

30. Two-sample $T^2$ uses pooled estimate:
  1. $\mathbf S_p=[(n_1-1)\mathbf S_1+(n_2-1)\mathbf S_2]/(n_1+n_2-2)$
  2. $(\mathbf S_1+\mathbf S_2)/2$
  3. $\mathbf S_1$
  4. None
Show Answer

A.

31. $\bar{\mathbf X}$ and $\mathbf S$ in MVN are:
  1. Independent
  2. Correlated
  3. Equal
  4. Sum to zero
Show Answer

A.

32. If $\mathbf X\sim N_p(\mu,\Sigma)$ then $\mathbf{AX}+\mathbf b\sim$
  1. $N_p(\mathbf A\mu+\mathbf b,\mathbf A\Sigma\mathbf A')$
  2. $N_p(\mu,\Sigma)$
  3. Cauchy
  4. $\chi^2$
Show Answer

A.

33. Equality of mean vectors of $k$ groups: test is:
  1. One-way MANOVA
  2. $T^2$
  3. PCA
  4. CCA
Show Answer

A.

34. For PCA, components corresponding to small eigenvalues are usually:
  1. Discarded
  2. Retained
  3. Used as main
  4. Plotted as outliers
Show Answer

A.

35. Scree plot helps decide:
  1. Number of PCs to retain
  2. Sample size
  3. Type of test
  4. Outliers
Show Answer

A.

36. Kaiser criterion in PCA: retain PCs with eigenvalues:
  1. $>1$
  2. $>0$
  3. $>0.5$
  4. $>2$
Show Answer

A. Kaiser's rule retains principal components with eigenvalue $>1$ (on the correlation matrix, whose eigenvalues average $1$).

37. Loading of variable $X_j$ on PC $Y_i$:
  1. $e_{ij}\sqrt{\lambda_i}/\sqrt{\sigma_{jj}}$
  2. $\lambda_i$
  3. $e_{ij}$
  4. $\sqrt{\lambda_i}$
Show Answer

A.

38. Discriminant function for two groups assigns to group 1 if:
  1. $a'x\ge \frac{1}{2}a'(\mu_1+\mu_2)$
  2. $a'x\le 0$
  3. $x\ge \mu_1$
  4. None
Show Answer

A.

39. Quadratic discriminant analysis is used when:
  1. Covariance matrices differ across groups
  2. Sample sizes equal
  3. Means equal
  4. Always
Show Answer

A.

40. Cross-validation in discriminant analysis is used to estimate:
  1. Variance
  2. Misclassification rate
  3. Mean
  4. None
Show Answer

B.

41. Box's M test checks:
  1. Equality of mean vectors
  2. Equality of covariance matrices
  3. MVN assumption
  4. Independence
Show Answer

B.

42. Roy's largest root statistic is for:
  1. Univariate test
  2. MANOVA
  3. Linear regression
  4. None
Show Answer

B.

43. Pillai's trace, Wilks' Lambda, Hotelling–Lawley, Roy — these are:
  1. MANOVA test statistics
  2. Goodness-of-fit measures
  3. Distribution functions
  4. Sampling designs
Show Answer

A.

44. For two-sample $T^2$, equal covariance assumption is:
  1. Required
  2. Optional
  3. Same as $\mu$
  4. None
Show Answer

A.

45. Confidence region for $\mu$ from MVN sample is:
  1. Ellipsoid
  2. Hypercube
  3. Half-space
  4. Line
Show Answer

A.

46. Sum of all eigenvalues of correlation matrix equals:
  1. $p$
  2. 1
  3. $n$
  4. $p(p-1)/2$
Show Answer

A. A $p\times p$ correlation matrix has trace $p$, so its eigenvalues sum to $p$.

47. The first eigenvector in PCA corresponds to:
  1. Smallest eigenvalue
  2. Largest eigenvalue
  3. Mean
  4. Median
Show Answer

B.

48. Two canonical variates on different sides are:
  1. Correlated only for matching pair
  2. All correlated
  3. None correlated
  4. $=$ correlation
Show Answer

A.

49. The test $H_0:\rho_1=\cdots=\rho_k=0$ in CCA uses:
  1. Wilks' $\Lambda$
  2. $z$ test
  3. Sign test
  4. $t$
Show Answer

A.

50. Mahalanobis $D^2$ was developed by:
  1. P. C. Mahalanobis
  2. C. R. Rao
  3. R. A. Fisher
  4. R. C. Bose
Show Answer

A.

9

Unit 9 — Stochastic Processes (50 MCQs)

Markov chains, Chapman–Kolmogorov, random walks, Poisson processes, queues.

1. A Markov chain has the property:
  1. Future depends on past given present
  2. Future independent of past given present
  3. Memory of all past states
  4. Stationary in time
Show Answer

B.

2. Transition probability matrix is:
  1. Symmetric
  2. Row stochastic
  3. Column stochastic
  4. Diagonal
Show Answer

B.

3. Chapman–Kolmogorov: $P^{m+n}=$
  1. $P^m P^n$
  2. $P^m+P^n$
  3. $P$
  4. $P^{mn}$
Show Answer

A.

4. Stationary distribution $\pi$ satisfies:
  1. $\pi P=\pi$
  2. $P\pi=\pi$
  3. $\pi=0$
  4. $\pi=1$
Show Answer

A.

5. For finite irreducible aperiodic chain, stationary distribution is:
  1. Unique and positive
  2. Multiple
  3. Zero
  4. Doesn't exist
Show Answer

A.

6. A state is recurrent if return prob equals:
  1. 1
  2. 0
  3. 1/2
  4. $\pi$
Show Answer

A.

7. A state is positive recurrent if:
  1. Mean return time finite
  2. Recurrent only
  3. Period 1
  4. Mean return time infinite
Show Answer

A.

8. Symmetric simple random walk on $\mathbb Z$ is:
  1. Transient
  2. Positive recurrent
  3. Null recurrent
  4. Ergodic
Show Answer

C. The symmetric random walk on $\mathbb Z$ returns to the origin with probability 1 but has infinite expected return time $-$ null recurrent.

9. Random walk on $\mathbb Z^d$ for $d\ge 3$ is:
  1. Recurrent
  2. Transient
  3. Periodic
  4. None
Show Answer

B. P\'olya's theorem: the symmetric walk is recurrent in dimensions 1 and 2 but transient for $d\ge 3$.

10. Period of state $i$: $d(i)=$
  1. $\gcd\{n\ge 1: p_{ii}^{(n)}>0\}$
  2. $\text{lcm}$
  3. $\min$
  4. $\max$
Show Answer

A.

11. Aperiodic state has period:
  1. 0
  2. 1
  3. 2
  4. $\infty$
Show Answer

B.

12. Gambler's ruin with fair coin, starting $i$ out of $N$: $P(\text{ruin})=$
  1. $(N-i)/N$
  2. $i/N$
  3. 1/2
  4. 0
Show Answer

A. In a fair game the ruin probability is $(N-i)/N$, linear in the distance from the target $N$.

13. Expected duration in fair gambler's ruin from $i$:
  1. $i(N-i)$
  2. $iN$
  3. $N^2$
  4. $\infty$
Show Answer

A.

14. Poisson process has:
  1. Stationary independent increments
  2. Dependent increments
  3. Bounded paths
  4. Symmetric
Show Answer

A.

15. Mean of Poisson process at time $t$:
  1. $\lambda$
  2. $\lambda t$
  3. $t$
  4. 0
Show Answer

B.

16. Variance of Poisson process at $t$:
  1. $\lambda t$
  2. $\lambda^2 t^2$
  3. $\lambda$
  4. $t$
Show Answer

A.

17. Inter-arrival times for Poisson process are:
  1. Uniform
  2. iid Exponential
  3. Gamma
  4. Normal
Show Answer

B.

18. Waiting time for $n$th arrival has distribution:
  1. Gamma$(n,\lambda)$
  2. Exp$(n\lambda)$
  3. $\chi^2$
  4. Uniform
Show Answer

A.

19. Memoryless property: $P(X>s+t\mid X>s)=$
  1. $P(X>t)$
  2. $P(X>s)$
  3. $P(X>s+t)$
  4. 0
Show Answer

A.

20. Birth and death process is a:
  1. Continuous-time Markov chain
  2. Discrete-time MC
  3. Renewal process
  4. Brownian motion
Show Answer

A.

21. Pure birth process (rate $\lambda$) is:
  1. Poisson
  2. Exponential
  3. Bernoulli
  4. Uniform
Show Answer

A.

22. M/M/1 queue notation: M means:
  1. Markov (memoryless)
  2. Mean
  3. Median
  4. Mathematical
Show Answer

A.

23. Traffic intensity $\rho=$
  1. $\lambda/\mu$
  2. $\mu/\lambda$
  3. $\lambda\mu$
  4. 1
Show Answer

A.

24. M/M/1 stationary distribution exists iff:
  1. $\rho>1$
  2. $\rho<1$
  3. $\rho=1$
  4. Always
Show Answer

B. M/M/1 has a stationary distribution iff the traffic intensity $\rho=\lambda/\mu<1$ (arrivals slower than service).

25. Stationary distribution of M/M/1: $\pi_n=$
  1. $(1-\rho)\rho^n$
  2. $\rho^n$
  3. $1/n$
  4. $e^{-\rho}\rho^n/n!$
Show Answer

A. The stationary queue length is geometric: $\pi_n=(1-\rho)\rho^n$.

26. Average number in M/M/1 system:
  1. $\rho/(1-\rho)$
  2. $\rho$
  3. $\lambda$
  4. $1/(1-\rho)$
Show Answer

A.

27. Average waiting in queue $W_q=$
  1. $\rho/(\mu-\lambda)$
  2. $\rho^2/(\mu-\lambda)$
  3. $1/\mu$
  4. $\lambda$
Show Answer

A.

28. Little's Law: $L=$
  1. $\lambda W$
  2. $W/\lambda$
  3. $\lambda+W$
  4. $\lambda^2 W$
Show Answer

A. Little's Law: $L=\lambda W$ $-$ mean number in system equals arrival rate times mean time in system.

29. Merging two independent Poisson processes with rates $\lambda_1,\lambda_2$ gives:
  1. Poisson with rate $\lambda_1+\lambda_2$
  2. Poisson with $\lambda_1\lambda_2$
  3. Not Poisson
  4. Exponential
Show Answer

A.

30. Thinning a Poisson process (each arrival kept w.p. $p$) gives:
  1. Poisson with $p\lambda$
  2. Binomial
  3. Geometric
  4. Not Poisson
Show Answer

A.

31. For an absorbing state $i$:
  1. $p_{ii}=1$
  2. $p_{ii}=0$
  3. $p_{ii}=1/2$
  4. Unknown
Show Answer

A.

32. For reversible chain: detailed balance is:
  1. $\pi_i p_{ij}=\pi_j p_{ji}$
  2. $\pi P=\pi$
  3. $p_{ij}=p_{ji}$
  4. None
Show Answer

A.

33. Stationary distribution of birth-death process:
  1. Always uniform
  2. $\pi_n=\pi_0\prod\lambda_k/\mu_{k+1}$
  3. Exponential
  4. Doesn't exist
Show Answer

B.

34. Mean return time in irreducible positive recurrent chain: $\mu_{jj}=$
  1. $1/\pi_j$
  2. $\pi_j$
  3. 1
  4. $\infty$
Show Answer

A. For an ergodic chain the mean recurrence time is the reciprocal of the stationary probability: $\mu_{jj}=1/\pi_j$.

35. Markov chain with two states $\{0,1\}$ and $p_{01}=p,p_{10}=q$: stationary $\pi_1=$
  1. $p/(p+q)$
  2. $q/(p+q)$
  3. 1/2
  4. $pq$
Show Answer

A.

36. Renewal process generalises:
  1. Poisson process
  2. Markov chain
  3. Brownian motion
  4. Random walk
Show Answer

A.

37. Time spent in state $i$ before transition in continuous-time MC:
  1. Exponential
  2. Gamma
  3. Uniform
  4. Discrete
Show Answer

A.

38. Compound Poisson process arises by:
  1. Summing iid jumps at Poisson times
  2. Two Poissons multiplied
  3. Time-reverse
  4. Squared Poisson
Show Answer

A.

39. Branching process expected size in generation $n$:
  1. $\mu^n$
  2. $\mu$
  3. $n\mu$
  4. 0
Show Answer

A.

40. Branching process: extinction certain iff:
  1. $\mu\le 1$
  2. $\mu>1$
  3. $\mu=0$
  4. Always
Show Answer

A. A branching process becomes extinct with probability 1 iff the mean number of offspring $\mu\le 1$ (subcritical or critical).

41. Brownian motion is a:
  1. Continuous Gaussian process with stationary independent increments
  2. Discrete
  3. Periodic
  4. Deterministic
Show Answer

A.

42. $E(B(t))=0$, $\text{Var}(B(t))=$
  1. $t$
  2. $t^2$
  3. $\sigma$
  4. 1
Show Answer

A.

43. For continuous-time MC, generator $Q$ has:
  1. Row sums = 0
  2. Row sums = 1
  3. Diagonal positive
  4. None
Show Answer

A.

44. For ergodic CTMC, stationary distribution $\pi$ satisfies:
  1. $\pi Q=0$
  2. $\pi P=\pi$ same
  3. $Q\pi=0$
  4. Both A and "$\sum\pi=1$"
Show Answer

D.

45. Order statistics of $n$ iid Uniform are conditioned arrival times of:
  1. Poisson process given $N(t)=n$
  2. Renewal
  3. BD process
  4. None
Show Answer

A.

46. M/M/c queue with $c$ servers stable iff:
  1. $\lambda<c\mu$
  2. $\lambda<\mu$
  3. $\rho=1$
  4. $\lambda c=\mu$
Show Answer

A.

47. For Yule (pure birth) with rate $\lambda i$: $E(N(t))=$
  1. $N_0 e^{\lambda t}$
  2. $\lambda t$
  3. $N_0$
  4. $e^{-\lambda t}$
Show Answer

A.

48. Communicating classes form an:
  1. Equivalence relation
  2. Partial order
  3. Total order
  4. None
Show Answer

A.

49. Recurrence is a:
  1. Property of individual state only
  2. Class property
  3. Both
  4. Neither
Show Answer

C. Property of class (all states share it).

50. The number of arrivals in disjoint intervals of Poisson process are:
  1. Dependent
  2. Independent
  3. Equal
  4. Always 1
Show Answer

B.

10

Unit 10 — Indian Statistical System & Research Methodology (50 MCQs)

MoSPI, NSC, NSO, census, Indian statisticians, R, LaTeX.

1. MoSPI stands for:
  1. Ministry of Statistics and Programme Implementation
  2. Ministry of Statistical Planning and Investment
  3. Ministry of Survey and Planning of India
  4. Multilateral Office of Statistical Planning India
Show Answer

A.

2. NSC stands for:
  1. National Statistical Commission
  2. National Survey Committee
  3. National Statistical Council
  4. National Sampling Centre
Show Answer

A.

3. NSO was formed in 2019 by merging:
  1. CSO and NSSO
  2. NSC and CSO
  3. NSSO and DGCIS
  4. RBI and NSC
Show Answer

A. The NSO was created in 2019 by merging the Central Statistics Office (CSO) and the National Sample Survey Office (NSSO).

4. Indian Statistical Institute was founded by:
  1. R. A. Fisher
  2. P. C. Mahalanobis
  3. C. R. Rao
  4. R. C. Bose
Show Answer

B.

5. Year of founding of ISI:
  1. 1925
  2. 1931
  3. 1947
  4. 1950
Show Answer

B.

6. National Sample Survey was started in:
  1. 1947
  2. 1950
  3. 1965
  4. 1980
Show Answer

B.

7. Mahalanobis Model influenced which Five Year Plan?
  1. First
  2. Second
  3. Fifth
  4. Tenth
Show Answer

B.

8. Rao–Cramer inequality is attributed to:
  1. R. A. Fisher
  2. C. R. Rao
  3. Harald Cramér
  4. Both B and C
Show Answer

D. The Cram\'er$-$Rao inequality is credited jointly to C. R. Rao and Harald Cram\'er, who derived it independently.

9. Rao–Blackwell theorem is named after:
  1. C. R. Rao and David Blackwell
  2. Rao and Bose
  3. Rao and Fisher
  4. Blackwell and Fisher
Show Answer

A.

10. Bose, Shrikhande, Parker disproved Euler's conjecture on:
  1. Latin squares
  2. BIBD
  3. Polynomials
  4. Goldbach
Show Answer

A.

11. S. N. Roy is known for:
  1. Multivariate analysis (Largest Root)
  2. Time series
  3. Sampling theory
  4. Statistical mechanics
Show Answer

A.

12. P. V. Sukhatme worked primarily on:
  1. Sampling in agriculture
  2. Time series
  3. Multivariate
  4. Bayesian
Show Answer

A.

13. Basu's theorem is due to:
  1. D. Basu
  2. R. R. Bahadur
  3. P. K. Sen
  4. Bose
Show Answer

A.

14. C. R. Rao received the International Prize in Statistics in:
  1. 2010
  2. 2015
  3. 2023
  4. 2020
Show Answer

C. C. R. Rao received the International Prize in Statistics in 2023, shortly before his death.

15. Census of India is conducted every:
  1. 5 years
  2. 10 years
  3. 20 years
  4. 1 year
Show Answer

B.

16. SRS in India stands for:
  1. Sample Registration System
  2. Simple Random Sampling
  3. Survey Reporting System
  4. State Registration System
Show Answer

A.

17. Census of India is conducted by:
  1. NSO
  2. Office of Registrar General & Census Commissioner
  3. RBI
  4. MoSPI
Show Answer

B.

18. NFHS stands for:
  1. National Family Health Survey
  2. National Forecast and Health System
  3. National Field Household Survey
  4. None
Show Answer

A.

19. PLFS stands for:
  1. Periodic Labour Force Survey
  2. Public Labour Force Survey
  3. Programme Labour Force System
  4. None
Show Answer

A.

20. ASI is conducted under which Act?
  1. Census Act 1948
  2. Collection of Statistics Act 2008
  3. RTI Act 2005
  4. IT Act 2000
Show Answer

B. The Annual Survey of Industries is conducted under the Collection of Statistics Act, 2008.

21. R was developed by:
  1. Ihaka and Gentleman
  2. Bell Labs
  3. Knuth
  4. Stallman
Show Answer

A.

22. In R, assignment operator is:
  1. $=$
  2. $<-$
  3. $\rightarrow$
  4. All of the above
Show Answer

D. R accepts <-, = and -> for assignment (<- is idiomatic).

23. Matrix multiplication in R uses:
  1. $*$
  2. %*%
  3. $\times$
  4. matmul()
Show Answer

B.

24. Inverse of matrix in R:
  1. inv(A)
  2. solve(A)
  3. 1/A
  4. A^(-1)
Show Answer

B.

25. Eigenvalues in R:
  1. eigen(A)$values
  2. eig(A)
  3. spec(A)
  4. roots(A)
Show Answer

A.

26. To draw $n$ samples from standard normal in R:
  1. rnorm(n)
  2. dnorm(n)
  3. pnorm(n)
  4. qnorm(n)
Show Answer

A.

27. CDF of normal in R is:
  1. pnorm()
  2. dnorm()
  3. rnorm()
  4. qnorm()
Show Answer

A.

28. Density of $\chi^2$ in R:
  1. dchisq()
  2. pchisq()
  3. rchisq()
  4. qchisq()
Show Answer

A.

29. NA in R represents:
  1. Not Available (missing)
  2. Negative Argument
  3. New Array
  4. Null Allocation
Show Answer

A.

30. mean(x, na.rm=TRUE) does:
  1. Includes NA
  2. Excludes NA when computing
  3. Returns NA
  4. Error
Show Answer

B.

31. In R, factor is used for:
  1. Numeric data
  2. Categorical data
  3. Text only
  4. Lists
Show Answer

B.

32. To get summary stats of dataframe:
  1. summary(df)
  2. desc(df)
  3. info(df)
  4. describe(df)
Show Answer

A.

33. To read CSV in R:
  1. read.csv()
  2. read.table()
  3. readLines()
  4. open.csv()
Show Answer

A.

34. plot(x,y) in R creates:
  1. Histogram
  2. Scatter plot
  3. Bar plot
  4. Boxplot
Show Answer

B.

35. hist(x) plots:
  1. Histogram
  2. Density
  3. Boxplot
  4. Scatter
Show Answer

A.

36. par(mfrow=c(2,2)) creates:
  1. 2x2 plot grid
  2. 2 plots
  3. 4 windows
  4. One plot
Show Answer

A.

37. apply(M,1,sum) gives:
  1. Row sums
  2. Column sums
  3. Total sum
  4. Diagonal
Show Answer

A.

38. To define a function in R: keyword is:
  1. def
  2. function
  3. lambda
  4. fun
Show Answer

B.

39. LaTeX is a typesetting system created by:
  1. Donald Knuth
  2. Leslie Lamport
  3. Richard Stallman
  4. Linus Torvalds
Show Answer

B. (TeX by Knuth; LaTeX by Lamport)

40. TeX was originally created by:
  1. Knuth
  2. Lamport
  3. Stallman
  4. Gentleman
Show Answer

A.

41. In LaTeX, inline math is delimited by:
  1. \[ \]
  2. $ $
  3. {}
  4. ""
Show Answer

B.

42. Display math in LaTeX:
  1. \[ ... \]
  2. $ ... $
  3. { ... }
  4. none
Show Answer

A.

43. \frac{a}{b} renders:
  1. $a/b$ as fraction
  2. Just text
  3. $\sqrt{a/b}$
  4. Subscript
Show Answer

A.

44. Subscript in LaTeX:
  1. x_i
  2. x^i
  3. x{i}
  4. x[i]
Show Answer

A.

45. The package needed for math symbols:
  1. amsmath
  2. graphicx
  3. geometry
  4. fontspec
Show Answer

A.

46. Indian Statistical Service is under:
  1. MoSPI
  2. UGC
  3. RBI
  4. SEBI
Show Answer

A.

47. CSO compiled traditionally:
  1. National Accounts, IIP
  2. Census
  3. Only PLFS
  4. RBI data
Show Answer

A.

48. The base year for current GDP series (as of 2024) is:
  1. 2004-05
  2. 2011-12
  3. 2018-19
  4. 2020-21
Show Answer

B. India's current national-accounts / GDP series uses base year 2011-12.

49. R Markdown combines:
  1. R code with Markdown text
  2. R with LaTeX only
  3. HTML only
  4. Word and R
Show Answer

A.

50. Overleaf is:
  1. Online LaTeX editor
  2. Statistical software
  3. Database
  4. Word processor for Indian languages
Show Answer

A.