List of top Questions asked in GATE ST

Let a random variable \( X \) follow a distribution with density \( f \in \{f_0, f_1\} \), where
\[ f_0(x) = \begin{cases} 1 & \text{if } 0 \leq x \leq 1 \\ 0 & \text{otherwise}, \end{cases} \] \[ f_1(x) = \begin{cases} 1 & \text{if } 1 \leq x \leq 2 \\ 0 & \text{otherwise}. \end{cases} \] Let \( \phi \) be a most powerful test of level 0.05 for testing \( H_0: f = f_0 \) against \( H_1: f = f_1 \) based on \( X \). Then which one of the following options is necessarily correct?

Let \( \{X_n\}_{n \geq 1} \) be a sequence of i.i.d. random variables with the common probability density function \[ f(x) = \frac{1}{\pi(1 + x^2)}, \quad -\infty<x<\infty. \] Define \[ Y_n = \frac{1}{2} + \frac{1}{\pi} \tan^{-1}(X_n) { for } n = 1, 2, \dots. \] Then which one of the following options is correct?

Let \( X \) be a random variable having the Poisson distribution with mean \( \log_e 2 \). Then \( E\left( e^{(\log_e 3)X} \right) \) equals:

Let \( S = \{(x, y, z) \in \mathbb{R}^3 \setminus \{(0,0,0)\} : z = -(x + y)\} \). Denote:

\[ S^\perp = \{(p, q, r) \in \mathbb{R}^3 : px + qy + rz = 0 \text{ for all } (x, y, z) \in S\}. \]

Then which one of the following options is correct?

Let \( \Phi(.) \) denote the cumulative distribution function of a standard normal random variable. If the random variable \( X \) has the cumulative distribution function \[ F(x) = \begin{cases} \Phi(x) & \text{if } x < -1, \\ \Phi(x + 1) & \text{if } x \geq -1, \end{cases} \] then which one of the following statements is true?

Let \( X \) be a random variable with probability density function \[ f(x) = \begin{cases} \alpha \lambda x^{\alpha - 1} e^{-\lambda x^\alpha} & \text{if } x > 0, \\ 0 & \text{otherwise} \end{cases} \] where \( \alpha > 0 \) and \( \lambda > 0 \). If the median of \( X \) is 1 and the third quantile is 2, then \( (\alpha, \lambda) \) equals:

Suppose that \( X \) has the probability density function \[ f(x) = \begin{cases} \frac{\lambda^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e^{-\lambda x} & \text{if } x > 0, \\ 0 & \text{otherwise}, \end{cases} \] where \( \alpha > 0 \) and \( \lambda > 0 \). \text{Which one of the following statements is NOT true?}

Let \( (X, Y) \) have joint probability density function \[ f(x, y) = \begin{cases} 8xy & \text{if } 0 < x < y < 1, \\ 0 & \text{otherwise.} \end{cases} \] If \( E(X \mid Y = y_0) = \frac{1}{2} \), then \( y_0 \) equals

Suppose that there are 5 boxes, each containing 3 blue pens, 1 red pen and 2 black pens. One pen is drawn at random from each of these 5 boxes. If the random variable \( X_1 \) denotes the total number of blue pens drawn and the random variable \( X_2 \) denotes the total number of red pens drawn, then \( P(X_1 = 2, X_2 = 1) \) equals:

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?

Consider the following regression model \[ y_k = \alpha_0 + \alpha_1 \log k + \epsilon_k, \quad k = 1, 2, \dots, n, \] where \( \epsilon_k \) are independent and identically distributed random variables each having probability density function \( f(x) = \frac{1}{2} e^{-|x|}, \ x \in \mathbb{R}. \) 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 \( \{X_n\}_{n \geq 1} \) be a Markov chain with state space \( \{1, 2, 3\} \) and transition probability matrix \[ P = \begin{bmatrix} \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ 0 & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \] Then \( P(X_2 = 1 \mid X_1 = 1, X_3 = 2) \) (rounded off to two decimal places) equals:

Suppose that \( (X_1, X_2, X_3) \) has a \( N_3(\mu, \Sigma) \) distribution with \[ \mu = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \quad \text{and} \quad \Sigma = \begin{pmatrix} 2 & 2 & 1 \\ 2 & 5 & 1 \\ 1 & 1 & 1 \end{pmatrix}. \] Given that \( \Phi(-0.5) = 0.3085 \), where \( \Phi(.) \) denotes the cumulative distribution function of a standard normal random variable, \[ P\left( (X_1 - 2X_2 + 2X_3)^2 < \frac{7}{2} \right) \text{ (rounded off to two decimal places) equals ...............} \]

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