List of top Statistics Questions asked in GATE Statistics

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 \( X_1, X_2, ...., X_{10} \) 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 \( A \) 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 \( f: \mathbb{R}^2 ⇒ \mathbb{R} \) 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 \( \{X_n\}_{n \geq 1} \) 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 \( A \) 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 ................

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?

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?