List of top Questions asked in GATE ST- 2026

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 \(X_1\) 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?

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 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 \(X_1,X_2,\ldots,X_{10}\) 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).

Consider the following two subspaces of \(\mathbb{R}^4\) \[ W_1=\{(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0\} \] \[ W_2=\{(x_1,x_2,x_3,x_4):x_1+2x_2+3x_3+4x_4=0\}. \] Which of the following statements is correct?

Let \(U_1,U_2,U_3\) be three independent exponential random variables such that \(U_k\) has the following probability density function
\[ f_k(x)=\begin{cases}ke^{-kx}&\text{if }x>0\\0&\text{otherwise,}\end{cases}\quad k=1,2,3. \]
Let
\[ X=\min\{U_1,U_3\}\quad\text{and}\quad Y=\min\{U_2,U_3\}. \]
Then \(P(X=Y)\) equals ______ (rounded 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 \(x_1,x_2,\ldots,x_n\) (\(n\ge2\)) 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).

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

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?

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).

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?

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 \(\{X_k\}_{k\geq1}\) 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?

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).

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?

Consider the polynomial \[ p(t)=t(t-1)(t-2) \] and define \[ F(x)=\int_{1/2}^{x}\frac{1}{p(t)}\,dt,\qquad x\in\left[\frac{1}{2},1\right). \] Which of the following statements is correct?

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?

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?

Let \(x_1\in(0,4)\) and consider the sequence \(\{x_n\}_{n\geq1}\) defined iteratively by
\[ x_{n+1}=2-(4-x_n)^{\frac{1}{2}},\quad n\geq1. \]
Consider the following statements:
(I) \(\{x_n\}\) converges to \(0\).
(II) \(\left\{\dfrac{x_{n+1}}{x_n}\right\}\) converges to \(\dfrac{1}{4}\).

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

Let \(f:(-1,1)\to\mathbb{R}\) be a differentiable function. Consider the following statements:
(I) Suppose \(f(0)\geq0\), and \(f'(x)>0\) whenever \(f(x)=0\), for any \(x\geq0\). Then \(f(x)>0\), for any \(x>0\).
(II) Suppose \(f(0)\leq0\), and \(f'(x)>0\) whenever \(f(x)=0\), for any \(x\leq0\). Then \(f(x)<0\), for any \(x<0\).
Which of the following statements is correct?

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?