List of top Questions asked in GATE Statistics

Consider the following regression model \[ y_t = \alpha_0 + \alpha_1 t + \alpha_2 t^2 + \epsilon_t, t = 1, 2, ...., 100, \] where $\alpha_0, \alpha_1, \alpha_2$ are unknown parameters and $\epsilon_t$'s are independent and identically distributed random variables each having $N(\mu, 1)$ distribution with $\mu \in \mathbb{R$ unknown. Then which one of the following statements is/are true?}

Consider the probability space \( (\Omega, \mathcal{G}, P) \), where \( \Omega = \{1, 2, 3, 4\} \), \[ \mathcal{G} = \{\emptyset, \Omega, \{4\}, \{2, 3\}, \{1, 4\}, \{1, 2, 3\}, \{2, 3, 4\}\}, \] and \( P(\{1\}) = \frac{1}{4} \). Let \( X \) be the random variable defined on the above probability space as \[ X(1) = 1, X(2) = X(3) = 2, X(4) = 3. \] If \( P(X \leq 2) = \frac{3}{4} \), then \( P(\{1, 4\}) \) (rounded off to two decimal places) equals ................

Consider a birth-death process on the state space \( \{ 0, 1, 2, 3 \} \). The birth rates are given by \( \lambda_0 = 1, \lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 0 \). The death rates are given by \( \mu_0 = 0, \mu_1 = 1, \mu_2 = 1, \mu_3 = 1 \). If \( [\pi_0, \pi_1, \pi_2, \pi_3] \) is the unique stationary distribution, then \( \pi_0 + 2\pi_1 + 3\pi_2 + 4\pi_3 \) (rounded off to two decimal places) equals:

Let \( \{-1, -\frac{1}{2}, 1, \frac{5}{2}, 3\} \) be a realization of a random sample of size 5 from a population having \( N \left(\frac{1}{2}, \sigma^2 \right) \) distribution, where \( \sigma>0 \) is an unknown parameter. Let \( T \) be an unbiased estimator of \( \sigma^2 \) whose variance attains the Cramer-Rao lower bound. Then based on the above data, the realized value of \( T \) (rounded off to two decimal places) equals ______________.

For a given real number \( a \), let \( a^+ = \max\{a, 0\} \) and \( a^- = \max\{-a, 0\} \). If \( \{x_n\}_{n \geq 1} \) is a sequence of real numbers, then which of the following statements is/are true?

Let \( X \) be a positive valued continuous random variable with finite mean. If \( Y = \lfloor X \rfloor \), the largest integer less than or equal to \( X \), then which of the following statements is/are true?

For any subset \( U \) of \( \mathbb{R}^n \), let \( L(U) \) denote the span of \( U \). For any two subsets \( T \) and \( S \) of \( \mathbb{R}^n \), which one of the following statements is NOT true?

Let \( f \) be a continuous function from \( [0, 1] \) to the set of all real numbers. Then which one of the following statements is NOT true?

Which of the following sets is/are countable?

Suppose that \( X_1, X_2, ...., X_{10} \) are independent and identically distributed random vectors each having \( N_2(\mu, \Sigma) \) distribution, where \( \Sigma \) is non-singular. If \[ U = \frac{1}{1 + (\bar{X} - \mu)^T \Sigma^{-1} (\bar{X} - \mu)} \] \text{where \( \bar{X} = \frac{1}{10} \sum_{i=1}^{10} X_i \), then the value of} \[ \log_e P(U \leq \frac{1}{2}) \text{ equals:} \]

Let \( A \) be an \( n \times n \) real matrix. Consider the following statements.
(I) If \( A \) is symmetric, then there exists \( c \geq 0 \) such that \( A + c I_n \) is symmetric and positive definite, where \( I_n \) is the \( n \times n \) identity matrix.
(II) If \( A \) is symmetric and positive definite, then there exists a symmetric and positive definite matrix \( B \) such that \( A = B^2 \).
\text{Which of the above statements is/are true?}

Let \( X_1, X_2, ...., X_{10} \) be a random sample of size 10 from a population having \( N(0, \theta^2) \) distribution, where \( \theta>0 \) is an unknown parameter. Let \( T = \frac{1}{10} \sum_{i=1}^{10} X_i^2 \). If the mean square error of \( cT \) (for \( c>0 \)), as an estimator of \( \theta^2 \), is minimized at \( c = c_0 \), then the value of \( c_0 \) equals

Let \( f: \mathbb{R}^2 ⇒ \mathbb{R} \) be defined by \( f(x, y) = xy \). Then the maximum value (rounded off to two decimal places) of \( f \) on the ellipse \( x^2 + 2y^2 = 1 \) equals ................

Let \( A \) be a \( 2 \times 2 \) real matrix such that \( AB = BA \) for all \( 2 \times 2 \) real matrices \( B \). If the trace of \( A \) equals 5, then the determinant of \( A \) (rounded off to two decimal places) equals ................

Two defective bulbs are present in a set of five bulbs. To remove the two defective bulbs, the bulbs are chosen randomly one by one and tested. If \( X \) denotes the minimum number of bulbs that must be tested to find out the two defective bulbs, then \( P(X = 3) \) (rounded off to two decimal places) equals:

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables, each having mean 4 and variance 9. If \( Y_n = \frac{1}{n} \sum_{i=1}^{n} X_i \) for \( n \geq 1 \), then \[ \lim_{n \to \infty} E \left[ \left( \frac{Y_n - 4}{\sqrt{n}} \right)^2 \right] \text{ (in integer) equals:} \]

Let \( X_1, X_2, ...., X_n \) be a random sample of size \( n \) from a population having uniform distribution over the interval \( \left( \frac{1}{3}, \theta \right) \), where \( \theta>\frac{1}{3} \) is an unknown parameter. If \( Y = \max \{ X_1, X_2, ...., X_n \} \), then which one of the following statements is true?

Suppose that \( X_1, X_2, ...., X_n, Y_1, Y_2, ...., Y_n \) are independent and identically distributed random vectors each having \( N_p(\mu, \Sigma) \) distribution, where \( \Sigma \) is non-singular, \( p>1 \) and \( n>1 \). If \( \overline{X} = \frac{1}{n} \sum_{i=1}^n X_i \) and \( \overline{Y} = \frac{1}{n} \sum_{i=1}^n Y_i \), then which one of the following statements is true?

Suppose that $x$ is an observed sample of size 1 from a population with probability density function $f(.)$. Based on $x$, consider testing \[ H_0: f(y) = \frac{1}{\sqrt{2\pi}} e^{-y^2/2}, y \in \mathbb{R} \text{against} H_1: f(y) = \frac{1}{2} e^{-|y|}, y \in \mathbb{R}. \] Then which one of the following statements is true?

Let \( X \) be a random variable having Poisson distribution with mean \( \lambda>0 \). Then \( E \left( \frac{1}{X+1} \mid X>0 \right) \) equals:

Let \( \{X_n\}_{n \geq 1} \) and \( \{Y_n\}_{n \geq 1} \) be two sequences of random variables and \( X \) and \( Y \) be two random variables, all of them defined on the same probability space. Which one of the following statements is true?

The area of the region bounded by the parabola \( x = -y^2 \) and the line \( y = x + 2 \) equals

Let \( A \) be a \( 3 \times 3 \) real matrix having eigenvalues \( 1, 0, \) and \( -1 \). If \( B = A^2 + 2A + I_3 \), where \( I_3 \) is the \( 3 \times 3 \) identity matrix, then which one of the following statements is true?

Based only on the truth of the statement ‘Some humans are intelligent’, which one of the following options can be logically inferred with certainty?