List of top Statistics Questions asked in GATE ST

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 \(X\) 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 \((\Omega,\mathcal F,P)\) 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 \(X\) be a single observation from a distribution having the probability density function
\[f_\theta(x)=\begin{cases}1 & \text{if } \theta<x<\theta+1\\0 & \text{otherwise,}\end{cases}\]
where \(\theta\in(-\infty,\infty)\). For testing \(H_0:\theta\le0\) against \(H_1:\theta>1\), let \(\beta\) be the power of the uniformly most powerful test of level \(0.05\). Then \(225\beta\) equals ______ (answer in integer).

Let \(X_1,X_2,\ldots,X_{10}\) be a random sample from the following probability density function
\[f(x)=\begin{cases}2(x-\mu)e^{-(x-\mu)^2} & \text{if } x>\mu\\0 & \text{otherwise,}\end{cases}\]
where \(\mu\in(-\infty,\infty)\) is an unknown parameter. It is given that the observed value of \(\min\{X_1,X_2,\ldots,X_{10}\}\) is \(1\). Using the pivot \(\min\{X_1,X_2,\ldots,X_{10}\}-\mu\), suppose a 95% confidence interval of \(\mu\) is of the form \((c,1)\), then \(c\) equals ______ (rounded off to two decimal places).

Let \((X_1,X_2,X_3)^T\) follow a trivariate normal distribution with mean vector \(\mu\) and covariance matrix \(\Sigma\) given by
\[\mu=\begin{pmatrix}1\\1\\1\end{pmatrix}\quad\text{and}\quad \Sigma=\begin{pmatrix}2&1&1\\1&2&1\\1&1&2\end{pmatrix}.\]
Then \(\text{Var}(X_1\mid X_2=1,X_3=-1)\) equals ______ (rounded off to two decimal places).

Let \(x_1,x_2,\ldots,x_n\) (\(n\ge2\)) be the observed values of a random sample from the following probability density function
\[f(x)=\begin{cases}\dfrac{\lambda^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\lambda x} & \text{if } x>0\\0 & \text{otherwise,}\end{cases}\]
where \(\alpha\in(0,\infty)\) and \(\lambda\in(0,\infty)\) are unknown parameters. If
\[\frac{x_1+x_2+\cdots+x_n}{n}=2 \quad\text{and}\quad \frac{x_1^2+x_2^2+\cdots+x_n^2}{n}=5,\]
then the method of moments estimate of \(\alpha\) equals ______ (answer in integer).

Let \(\{W(t):t\ge0\}\) be a standard Brownian motion, with \(W(0)=0\). Define
\[Z_1=W(1)+W(2)\quad\text{and}\quad Z_2=W(2)+W(3).\]
Let \(\rho\) be the correlation coefficient between \(Z_1\) and \(Z_2\). Then the value of \(10\rho\) is ______ (round off to two decimal places).

Consider the function \(f:\mathbb{R}^2 \to \mathbb{R}\) defined by
\[ f(x_1,x_2)=2x_1^4+x_2^2+x_2x_1^2. \]
Which of the following statements is correct?

Let \(X\) be a random variable such that
\[ P(X=i)=2P(X=i-1), \quad i=2,3,\ldots,n,\ n\geq7, \]
and \(\displaystyle\sum_{i=1}^{n}P(X=i)=1\). Then \((2^n-1)P(X=7)\) equals ________ (answer in integer).

Let \(X\) and \(Y\) be two continuous random variables having the following joint probability density function
\[ f(x,y)=\begin{cases} x+y & \text{if } 0<x<1,\ 0<y<1 \\ 0 & \text{otherwise}. \end{cases} \]
Then \(72\big(\mathrm{Var}(X)+\mathrm{Var}(Y)\big)\) equals ________ (answer in integer).

Consider the system of linear equations
\[ x+y+z=1, \]\[ 2x+y+3z=6, \]\[ 3x+2y+kz=k+1. \]If the above system has no solution, then the value of \(k\) equals _______ (answer in integer).

Consider the subspace
\[ U = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1+x_2+x_4=0,\ x_3=0\}. \]
Let \(U^{\perp}\) be its orthogonal complement. The value of \(\dim(U^{\perp})\) equals ________ (answer in integer).

Let \(X\) be a continuous random variable having distribution function \(F\). If
\[ Y=-3\ln F(X), \]
then \(E(Y)\) equals ________ (answer in integer).

The value of
\[ \lim_{n\to\infty} n\int_{1-\frac{1}{2n}}^{1+\frac{1}{2n}} e^{t^2-1}\,dt \]equals _______ (answer in integer).

Let \(X_1, X_2\) and \(X_3\) be three independent random variables such that \(X_k\) \((k=1,2,3)\) has the following probability density function
\[ f_k(x)=\begin{cases} k\,e^{-kx} & \text{if } x>0 \\ 0 & \text{otherwise}. \end{cases} \]
Let \(Y=\min\{X_1,X_2,X_3\}\). Then the value of \(E(3Y^2-Y)\) equals ________ (answer in integer).

For a random sample \(X_1,\ldots,X_n\), \(n\geq2\), from a population with distribution function \(F_X\), let \(Z_n(x)\) be the proportion of sample values less than or equal to \(x\), \(x\in\mathbb{R}\). Which of the following statements is/are true?

For testing a null hypothesis \(H_0\) against an alternative hypothesis \(H_1\) at level of significance \(\alpha \in (0,1)\), which of the following statements is correct?

Let \((X_1,Y_1),(X_2,Y_2),\ldots,(X_n,Y_n)\), \(n\geq2\), be a random sample from a continuous bivariate distribution with joint distribution function \(F_{X,Y}\). Further, \(F_X\) and \(F_Y\) are the marginal distribution functions of \(X\) and \(Y\), respectively. If
\[ F_{X,Y}(x,y)=F_X(x)F_Y(y), \quad \forall (x,y), \]then, for any two independent pairs \((X_i,Y_i)\) and \((X_j,Y_j)\),
\[ P\left[(X_i-X_j)(Y_i-Y_j)>0\right] \]equals

Let \(X_1\) and \(X_2\) be independent and identically distributed random variables following normal distribution with mean \(\theta\in(-\infty,\infty)\) and variance \(1\). Then which of the following estimators of their expected values attains the Cramer-Rao lower bound?

Let \(X_1,X_2\) be a random sample from the following probability density function
\[ f_\alpha(x)= \begin{cases} \alpha x^{\alpha-1}e^{-x^\alpha} & \text{if } x>0 \\ 0 & \text{otherwise,} \end{cases} \]
where \(\alpha\in(0,\infty)\) is an unknown parameter. Let
\[ X_{(1)}=\min\{X_1,X_2\} \quad\text{and}\quad X_{(2)}=\max\{X_1,X_2\}. \]
Then which of the following statements is correct?

Let \(X\) and \(Y\) be independent and identically distributed normal random variables. If
\[ P(X+2Y\le3)=P(2X-Y\ge4), \]
then \(E(X)\) is

Let \(X\) and \(Y\) be identically distributed random variables with variance \(\sigma^2\in(0,\infty)\). Then the correlation coefficient between \(X\) and \(Y\) is

Let \(X\) and \(Y\) be two independent discrete random variables such that the moment generating functions of \(X\) and \(X+Y\) are given by \[ M_X(t)=\frac{1+2e^{-t}+3e^{2t}}{6},\quad t\in\mathbb{R}, \] and \[ M_{X+Y}(t)=\frac{2+e^{t}+3e^{3t}}{6},\quad t\in\mathbb{R}, \] respectively. Then which of the following statements is correct?