List of top Statistics Questions asked in GATE Statistics

Let \( (X, Y)^T \) follow a bivariate normal distribution with \[ E(X) = 2, \, E(Y) = 3, \, {Var}(X) = 16, \, {Var}(Y) = 25, \, {Cov}(X, Y) = 14. \] Then \[ 2\pi \left( \Pr(X>2, Y>3) - \frac{1}{4} \right) \] equals _________ (rounded off to two decimal places).

Let \( X_1, X_2 \) be a random sample from a population having probability density function
\[ f_{\theta}(x) = \begin{cases} e^{(x-\theta)} & \text{if } -\infty < x \leq \theta, \\ 0 & \text{otherwise}, \end{cases} \] where \( \theta \in \mathbb{R} \) is an unknown parameter. Consider testing \( H_0: \theta \geq 0 \) against \( H_1: \theta < 0 \) at level \( \alpha = 0.09 \). Let \( \beta(\theta) \) denote the power function of a uniformly most powerful test. Then \( \beta(\log_e 0.36) \) equals ________ (rounded off to two decimal places).

Let \( X_1, X_2, \dots, X_7 \) be a random sample from a population having the probability density function \[ f(x) = \frac{1}{2} \lambda^3 x^2 e^{-\lambda x}, \quad x>0, \] where \( \lambda>0 \) is an unknown parameter. Let \( \hat{\lambda} \) be the maximum likelihood estimator of \( \lambda \), and \( E(\hat{\lambda} - \lambda) = \alpha \lambda \) be the corresponding bias, where \( \alpha \) is a real constant. Then the value of \( \frac{1}{\alpha} \) equals __________ (answer in integer).

Let \( \{ X_n \}_{n \geq 1} \) be a sequence of independent random variables with \[ \Pr(X_n = -\frac{1}{2^n}) = \Pr(X_n = \frac{1}{2^n}) = \frac{1}{2}, \quad \forall n \in \mathbb{N}. \] Suppose that \( \sum_{i=1}^{n} X_i \) converges to \( U \) as \( n \to \infty \). Then \( 6 \Pr(U \leq \frac{2}{3}) \) equals ___________ (answer in integer).

The service times (in minutes) at two petrol pumps \( P_1 \) and \( P_2 \) follow distributions with probability density functions \[ f_1(x) = \lambda e^{-\lambda x}, \quad x>0 \quad {and} \quad f_2(x) = \lambda^2 x e^{-\lambda x}, \quad x>0, \] respectively, where \( \lambda>0 \). For service, a customer chooses \( P_1 \) or \( P_2 \) randomly with equal probability. Suppose, the probability that the service time for the customer is more than one minute, is \( 2e^{-2} \). Then the value of \( \lambda \) equals _________ (answer in integer).

The moment generating functions of three independent random variables \( X, Y, Z \)  are respectively given as: \[ M_X(t) = \frac{1}{9}(2 + e^t)^2, \quad t \in \mathbb{R}, \] \[ M_Y(t) = e^{e^t - 1}, \quad t \in \mathbb{R}, \] \[ M_Z(t) = e^{2(e^t - 1)}, \quad t \in \mathbb{R}. \] Then \( 10 \cdot \Pr(X>Y + Z) \) equals __________ (rounded off to two decimal places).

Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be defined as \[ f(x, y) = x^2 y^2 + 8x - 4y. \] The number of saddle points of \( f \) is _________ (answer in integer).

For \( Y \in \mathbb{R}^n \), \( X \in \mathbb{R}^{n \times p} \), and \( \beta \in \mathbb{R}^p \), consider a regression model \[ Y = X \beta + \epsilon, \] where \( \epsilon \) has an \( n \)-dimensional multivariate normal distribution with zero mean vector and identity covariance matrix. Let \( I_p \) denote the identity matrix of order \( p \). For \( \lambda>0 \), let \[ \hat{\beta}_n = (X^T X + \lambda I_p)^{-1} X^T Y, \] be an estimator of \( \beta \). Then which of the following options is/are correct?

Let \( X = (X_1, X_2, X_3)^T \) be a 3-dimensional random vector having multivariate normal distribution with mean vector \( (0, 0, 0)^T \) and covariance matrix
\[ \Sigma = \begin{pmatrix} 4 & 0 & 0 <br> 0 & 9 & 0 <br> 0 & 0 & 4 \end{pmatrix}. \]
{Let } \( \alpha^T = (2, 0, -1) \) { and } \( \beta^T = (1, 1, 1) \). 
Then which of the following statements is/are correct?

Let \( X_1, X_2, X_3 \) be independent standard normal random variables, and let \[ Y_1 = X_1 - X_2, \quad Y_2 = X_1 + X_2 - 2X_3, \quad Y_3 = X_1 + X_2 + X_3. \] Then which of the following options is/are correct?

Consider the simple linear regression model \[ y_i = \alpha + \beta x_i + \epsilon_i, \quad i = 1, 2, \dots, 24, \] where \( \alpha \in \mathbb{R} \) and \( \beta \in \mathbb{R} \) are unknown parameters, the errors \( \epsilon_i \)'s are i.i.d. random variables having \( N(0, \sigma^2) \) distribution, where \( \sigma>0 \) is unknown. Suppose the following summary statistics are obtained from a data set of 24 observations \( (x_1, y_1), \dots, (x_{24}, y_{24}) \): \[ S_{xx} = \sum_{i=1}^{24} (x_i - \bar{x})^2 = 22.82, \quad S_{yy} = \sum_{i=1}^{24} (y_i - \bar{y})^2 = 43.62, \quad S_{xy} = \sum_{i=1}^{24} (x_i - \bar{x})(y_i - \bar{y}) = 15.48, \] where \( \bar{x} = \frac{1}{24} \sum_{i=1}^{24} x_i \) and \( \bar{y} = \frac{1}{24} \sum_{i=1}^{24} y_i \). Then, for testing \( H_0: \beta = 0 \) against \( H_1: \beta \neq 0 \), the value of the \( F \)-test statistic based on the least squares estimator of \( \beta \), whose distribution is \( F_{1,22} \), equals (rounded off to two decimal places):

Let \( X_1, X_2 \) be a random sample from a distribution having probability density function 

Let \( X \) and \( Y \) be discrete random variables with joint probability mass function \[ p_{X, Y}(m, n) = \frac{\lambda^n e^{-\lambda} 2^n m! (n - m)!}{n!}, \quad m = 0, \dots, n, \quad n = 0, 1, 2, \dots, \] where \( \lambda \) is a fixed positive real number. Then which one of the following options is correct?