List of top Statistics Questions on Statistics asked in GATE ST

Let \( X \) be a random variable with probability density function \[ f(x; \lambda) = \begin{cases} \frac{1}{\lambda} e^{- \frac{x}{\lambda}} & \text{if } x > 0, \\ 0 & \text{otherwise} \end{cases} \] where \( \lambda > 0 \) is an unknown parameter. Let \( Y_1, Y_2, \dots, Y_n \) be a random sample of size \( n \) from a population having the same distribution as \( X^2 \). If \( \overline{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i \), then which one of the following statements is true?

Let \( X_1, X_2, \dots, X_n \) be a random sample of size \( n \geq 2 \) from a population having probability density function \[ f(x; \theta) = \left\{ \begin{array}{ll} \frac{2}{\theta} \left( -\log_e x \right)^2 e^{-\left( \frac{\log_e x}{\theta} \right)^2} & \text{if } 0 < x < 1, \\ 0 & \text{otherwise} \end{array} \right. \] where \( \theta > 0 \) is an unknown parameter. Then which one of the following statements is true?

Suppose that \( X_1, X_2, \dots, X_n \) are independent and identically distributed random variables each having probability density function \( f(.) \) and median \( \theta \). We want to test \[ H_0: \theta = \theta_0 \quad \text{against} \quad H_1: \theta > \theta_0. \] Consider a test that rejects \( H_0 \) if \( S > c \) for some \( c \) depending on the size of the test, where \( S \) is the cardinality of the set \( \{i: X_i > \theta_0, 1 \leq i \leq n\} \). Then which one of the following statements is true?

Let \( \{W_t\}_{t \geq 0} \) be a standard Brownian motion. Then \( E(W_4^2 \mid W_2 = 2) \) (in integer) equals:

Let \( X \) be a random variable with probability density function \[ f(x) = \begin{cases} \frac{1}{x^2} & \text{if } x \geq 1, \\ 0 & \text{otherwise.} \end{cases} \] If \( Y = \log X \), then \( P(Y < 1 \mid Y < 2) \) equals

Let \( X_1, X_2, \dots, X_n \) be a random sample of size \( n \) from a population having probability density function \[ f(x; \mu) = \begin{cases} e^{-(x-\mu)} & \text{if } \mu \leq x < \infty \\ 0 & \text{otherwise} \end{cases} \] where \( \mu \in \mathbb{R} \) is an unknown parameter. If \( \hat{M} \) is the maximum likelihood estimator of the median of \( X_1 \), then which one of the following statements is true?

Let \( g(x) = f(x) + f(2 - x) \) for all \( x \in [0, 2] \), where \( f : [0, 2] \to \mathbb{R} \) is continuous on \( [0, 2] \) and twice differentiable on \( (0, 2) \). If \( g' \) denotes the first derivative of \( g \) and \( f'' \) denotes the second derivative of \( f \), then which of the following statements is NOT true?

Let \( X \) be a random variable with cumulative distribution function \[ F(x) = \begin{cases} 0 & \text{if } x < -1, \\ \frac{1}{4}(x + 1) & \text{if } -1 \leq x < 0, \\ \frac{1}{4}(x + 3) & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1 \end{cases} \] Which one of the following statements is true?

Let \( X_1, X_2, \dots, X_{10} \) be a random sample of size 10 from a \( N_3(\mu, \Sigma) \) distribution, where \( \mu \) and non-singular \( \Sigma \) are unknown parameters. If \[ \overline{X_1} = \frac{1}{5} \sum_{i=1}^{5} X_i, \quad \overline{X_2} = \frac{1}{5} \sum_{i=6}^{10} X_i, \] \[ S_1 = \frac{1}{4} \sum_{i=1}^{5} (X_i - \overline{X_1})(X_i - \overline{X_1})^T, \quad S_2 = \frac{1}{4} \sum_{i=6}^{10} (X_i - \overline{X_2})(X_i - \overline{X_2})^T, \] then which one of the following statements is NOT true?

Suppose that \( U \) and \( V \) are two independent and identically distributed random variables each having probability density function \[ f(x) = \begin{cases} \lambda^2 x e^{-\lambda x} & \text{if } x > 0 \\ 0 & \text{otherwise}, \end{cases} \] where \( \lambda > 0 \). Which of the following statements is/are true?

Let \( (X, Y) \) have joint probability mass function \[ p(x, y) = \begin{cases} \frac{e^{-2}}{x! (y - x)!} & \text{if } x = 0, 1, 2, \dots, y; \, y = 0, 1, 2, \dots \\ 0 & \text{otherwise}. \end{cases} \] Then which of the following statements is/are true?

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables with mean 0 and variance 1, all of them defined on the same probability space. For \( n = 1, 2, 3, \dots \), let \[ Y_n = \frac{1}{n} \left( X_1 X_2 + X_3 X_4 + \dots + X_{2n-1} X_{2n} \right). \] Then which one of the following statements is/are true?

Let X be a random sample of size 1 from a population with cumulative distribution function \[ F(x) = \begin{cases} 0 & \text{if } x < 0, \\ 1 - (1 - x)^\theta & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1 \end{cases} \] where \( \theta > 0 \) is an unknown parameter. To test \( H_0 : \theta = 1 \) against \( H_1 : \theta = 2 \), consider using the critical region \( \{ x \in \mathbb{R} : x < 0.5 \} \). If \( \alpha \) and \( \beta \) denote the level and power of the test, respectively, then \( \alpha + \beta \) (rounded off to two decimal places) equals:

Let \( \{0.13, 0.12, 0.78, 0.51\} \) be a realization of a random sample of size 4 from a population with cumulative distribution function \( F(.) \). Consider testing \( H_0: F = F_0 \) against \( H_1: F \neq F_0 \), where \[ F_0(x) = \begin{cases} 0 & \text{if } x < 0, \\ x & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1. \end{cases} \] Let \( D \) denote the Kolmogorov-Smirnov test statistic. If \( P(D > 0.669) = 0.01 \) under \( H_0 \) and \( \psi = \begin{cases} 1 & \text{if } H_0 \text{ is accepted at level 0.01}, \\ 0 & \text{otherwise} \end{cases} \)
then based on the given data, the observed value of } \( D + \psi \)
(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 \( X_1, X_2, ...., X_{10} \) 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 ................