List of practice Questions

Let \(A\in M_n(\mathbb{R})\) be an \(n\times n\) real matrix, \(n\geq2\). Consider the following two statements:
(I) If \(\lambda\in\mathbb{C}\) is an eigenvalue of \(A\), then its complex conjugate \(\bar\lambda\) is also an eigenvalue.
(II) If \(v=(v_1,v_2,\ldots,v_n)\in\mathbb{C}^n\) is an eigenvector corresponding to eigenvalue \(\lambda=x+iy\), \(y\neq0\), then
\[ \text{Re}(v)=(\text{Re}(v_1),\ldots,\text{Re}(v_n))\quad\text{and}\quad\text{Im}(v)=(\text{Im}(v_1),\ldots,\text{Im}(v_n)) \]
are linearly independent vectors over \(\mathbb{R}\).

Which of the following statements is correct?

Consider the system of equations
\[ x+2y-z=a \]
\[ x+y+3z=b \]
\[ 2x+3y+2z=c \]

Consider the following statements:
(I) For every \((a,b,c)\in\mathbb{R}^3\), the above system has a solution.
(II) For \((a,b,c)=(0,0,0)\), the solution set is given by
\[ \{(-7t,\ 4t,\ t):t\in\mathbb{R}\}. \]
Which of the following statements is correct?

Let \((\Omega,\mathcal F,P)\) be a probability space, where for \(A\subset\Omega\), \(A\neq\phi\), \(A\neq\Omega\),
\[ \mathcal F=\{\Omega,\phi,A,A^c\},\quad P(\Omega)=1,\ P(\phi)=0,\ P(A)=\frac{1}{2}=P(A^c). \]
Let \(X\) and \(Y\) be two random variables defined on \(\Omega\) as follows:
\[ X(\omega)=\begin{cases}1 & \text{if }\omega\in A\\ 0 & \text{if }\omega\in A^c,\end{cases} \quad\text{and}\quad Y(\omega)=\begin{cases}1 & \text{if }\omega\in A^c\\ 0 & \text{if }\omega\in A.\end{cases} \]
Then which of the following statements is correct?

Let \(X\) and \(Y\) be independent and identically distributed geometric random variables having the following probability mass function
\[ P(X=x)=p(1-p)^x, \quad x=0,1,2,\ldots, \]
where \(p\in(0,1)\). Then which of the following statements is correct?

Let \(X\) be a random variable with support \(S=\{0,1,2,\ldots\}\) and
\[ P(X\geq k+1\,|\,X\geq k)=p, \quad k\in S,\ \ 0<p<1. \]
Then which of the following statements is correct?

Let \(X_1,X_2,X_3\) be a random sample from a distribution having probability mass function
\[ f_{\theta}(x)=\begin{cases}\theta & \text{if } x=1\\ 1-\theta & \text{if } x=2\\ 0, & \text{otherwise},\end{cases} \]
where \(\theta\in\Theta=(0,1)\). Let \(\underline{X}=(X_1,X_2,X_3)\). Then which of the following is NOT a sufficient statistic for \(\theta\)?

Let \(X_1,X_2,\ldots,X_n\ (n\geq2)\) be a random sample from the probability density function \(f(x)\). Consider the following hypotheses:
\[ H_0: f(x)=\frac{1}{\sqrt{2\pi}}\,e^{-\frac{x^2}{2}};\quad -\infty<x<\infty \]
\[ H_1: f(x)=\frac{1}{2}\,e^{-|x|};\quad -\infty<x<\infty. \]
For testing \(H_0\) against \(H_1\), let \(R\) denote the critical region based on the likelihood ratio test having level \(0.05\). Then, for some constant \(c\), the region \(R\) is

Let \(X_1,X_2,\ldots,X_n\ (n>1)\) be a random sample from the following probability density function
\[ f_{\beta}(x)=\begin{cases}\beta e^{-x}(1-e^{-x})^{\beta-1} & \text{if } x>0\\ 0 & \text{otherwise},\end{cases} \]
where \(\beta>0\) is an unknown parameter. For testing the following hypotheses,
\[ H_0:\beta=1 \quad \text{against} \quad H_1:\beta>1, \]
at level \(\alpha\in(0,1)\), which of the following statements is correct?

Let \(X_1, X_2, \ldots, X_5\) be random observations from a continuous distribution. Let \(\theta_p\) be the \(p\)-th population quantile. Consider the following hypotheses
\[ H_0:\theta_{1/2}=2.5 \quad \text{against} \quad H_1:\theta_{1/2}>2.5. \]
Let \(X_{(r)}\) denote the \(r\)-th order statistic of the given observations. Then which of the following is a critical region of a level \(0.05\) test?

Suppose that \(X\) and \(Y\) are independent and identically distributed \(N_p(\mu, \Sigma)\) random vectors, where \(\mu \in \mathbb{R}^p\) and \(\Sigma\) is a positive definite matrix. Let \(\chi^2_m\) denote chi-square distribution with \(m\)-degrees of freedom. Then which of the following statements is correct?

Consider the multiple linear regression model
\[ Y_i = \beta_0+\beta_1 x_{i1}+\beta_2 x_{i2}+\beta_3 x_{i3}+\epsilon_i, \quad i=1,2,\ldots,31, \]
where \(\epsilon_i\) are iid \(N(0,1)\) variables. The \(F\)-test for testing significance of regression rejects \(H_0: \beta_1=\beta_2=\beta_3=0\) at \(5\%\) level. Given \[ F_{0.05;3,27}=2.96,\quad F_{0.05;3,30}=2.92,\quad F_{0.025;3,27}=4.01,\quad F_{0.025;3,30}=3.91. \] Then the value of \(R^2\) cannot be equal to

Consider the function \(f:\mathbb{R}^2\to\mathbb{R}\) defined by
\[ f(x,y)= \begin{cases} \dfrac{x^3+y^3}{\sqrt{x^2+2y^2}} & (x,y)\neq (0,0) \\ 0 & (x,y)=(0,0). \end{cases} \]
Which of the following statements is/are correct?

Consider the matrix
\[ A=\begin{pmatrix} \lambda_1 & 1 & 0 \\ 0 & \lambda_2 & 1 \\ 0 & 0 & \lambda_3 \end{pmatrix}, \qquad \lambda_1,\lambda_2,\lambda_3\in\mathbb{R}. \]
Which of the following statements is/are correct?

Consider the real symmetric matrix \(A=(a_{ij})\) given by \[ A=\begin{pmatrix}3 & 1 & 1\\ 1 & 0 & 2\\ 1 & 2 & 0\end{pmatrix}. \] Consider the set \[ S=\left\{x=(x_1,x_2,x_3)\in\mathbb{R}^3 : \sum_{j=1}^{3}\sum_{i=1}^{3}a_{ij}x_ix_j=1\right\}. \] Which of the following statements is/are correct?

Let \(X\) be a discrete random variable with support \(S=\{1,2,3,\ldots\}\) such that \(E(X^2)<\infty\). Let \(F\) be the distribution function of \(X\). Then which of the following statements is/are correct?

Let \(\{X_n\}_{n\geq1}\) be a sequence of random variables having the following probability mass function \[ P(X_n=x)=\frac{1}{5n}\left(1-\frac{1}{5n}\right)^{x},\quad x=0,1,2,\ldots;\ n\in\mathbb{N}. \] Define \(Z_n=\dfrac{X_n}{n}\), \(n\in\mathbb{N}\), and let \(V\) be a random variable. If \(Z_n\xrightarrow{d}V\), as \(n\to\infty\), then which of the following statements is/are correct?

Let \(\{X_k\}_{k\geq1}\) be a sequence of independent random variables such that \[ X_{2k-1}\sim \text{Bin}(1,\theta),\quad\text{and}\quad X_{2k}\sim \text{Bin}(1,1-\theta),\quad k=1,2,3,\ldots, \] where \(\theta\in(0,1)\). Let \(\{Y_k\}_{k\geq1}\) be another sequence of independent and identically distributed random variables such that \(Y_k\sim\text{Poisson}(\lambda)\), \(\lambda>0\). Define, for \(n\in\mathbb{N}\), \[ S_{2n}=\sum_{k=1}^{n}(X_{2k-1}-X_{2k}+1-2\theta),\quad W_n=\sum_{k=1}^{n}Y_k^2\quad\text{and}\quad \sigma_{2n}^2=2n\theta(1-\theta). \] Then which of the following statements is/are correct?

Let \(\{N(t);t\geq0\}\) be a homogeneous Poisson process with rate \(3\), and let \(T_1\) denote the first arrival time. Then which of the following statements is/are correct?

Let \(X_1, X_2, \ldots, X_n\) \((n \geq 2)\) be a random sample from the following probability density function
\[ f(x) = \frac{1}{2} e^{-|x-\mu|}, \quad -\infty < x < \infty, \] where \(\mu \in (-\infty, \infty)\) is an unknown parameter. Let \(\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i\) and \(\hat{\mu}\) denote the maximum likelihood estimator of \(\mu\), whenever it exists. Then which of the following statements is/are correct?

Let \(X\) be a single observation from a distribution having a probability density function \(f_\theta\), \(\theta \in \Theta\). For testing \(H_0: \theta = 1\) against \(H_1: \theta \neq 1\), under which of the following options a uniformly most powerful test of level \(0.05\) exists?

Suppose that \(X = (X_1, \ldots, X_p)^T\) follows \(N_p(\mu, \Sigma)\), where \(\mu \in \mathbb{R}^p\) and \(\Sigma\) is a positive definite matrix. Let \(A\) be a \(p \times p\) matrix such that \(A^T A = I_p\).
Let \(Y = (Y_1, \ldots, Y_p)^T = AX\). Then which of the following statements is/are correct?

Suppose that \(X = (X_1, \ldots, X_p)^T\) follows \(N_p(0, \Sigma)\), where \(\Sigma\) is a positive definite matrix. Then which of the following statements is/are correct?

Let \(Y = (Y_1, Y_2, Y_3)^T \sim N_3(0, I_3)\), where \(I_3\) denotes the identity matrix of order \(3\). Let
\[ A = \begin{pmatrix} \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{pmatrix} \quad \text{and} \quad B = I_3 - A. \] Let \(\chi_1^2\) denote chi-square distribution with \(1\) degree of freedom. Then which of the following statements is/are correct?

Let \(D=\{(x,y)\in\mathbb{R}^2:0<x^2+y^2\leq1\}\). For \(\alpha\geq0\), consider the integral
\[ I_\alpha=\iint_D \frac{1}{(x^2+y^2)^\alpha}\,dx\,dy. \]
Let
\[ N_0=\sup\{\alpha\geq0:I_\alpha<\infty\}. \]
Then the value of \(N_0\) equals ______ (answer in integer).

Let \(X\) follow \(N(3,1)\). Then the value of \(E\big(X^4(X-3)\big)\) equals ______ (answer in integer).