List of top Statistics Questions asked in GATE Statistics

Let \( U = \{(x, y) \in \mathbb{R}^2 : x + y \leq 2\} \). Define \( f: U \to \mathbb{R} \) by \[ f(x, y) = (x - 1)^4 + (y - 2)^4. \] The minimum value of \( f \) over \( U \) is: (answer in integer).

Consider the multi-linear regression model \[ y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + \beta_3 x_{3i} + \beta_4 x_{4i} + \epsilon_i, \quad i = 1, 2, \dots, 25, \] {where } \( \beta_i, i = 0, 1, 2, 3, 4 \) 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 that the value of the coefficient of determination \( R^2 \) is obtained as \( \frac{5}{6} \). Then the value of adjusted \( R^2 \) is _________ (rounded off to two decimal places).

Let \( (1, 3), (2, 4), (7, 8) \) be three independent observations. Then the sample Spearman rank correlation coefficient based on the above observations is ________ (rounded off to two decimal places).

Let \( X_1, \dots, X_5 \) be a random sample from \( N(\theta, 6) \), where \( \theta \in \mathbb{R} \), and let \( c(\theta) \) be the Cramer-Rao lower bound for the variances of unbiased estimators of \( \theta \) based on the above sample. Then \( 15 \cdot \inf_{\theta \in \mathbb{R}} c(\theta) \) equals _________ (answer in integer).

Let \( x_1 = 0, x_2 = 1, x_3 = 1, x_4 = 1, x_5 = 0 \) be observed values of a random sample of size 5 from \( {Bin}(1, \theta) \) distribution, where \( \theta \in (0, 0.7] \). Then the maximum likelihood estimate of \( \theta \) based on the above sample is ________ (rounded off to two decimal places).

Let \( \{ W(t) : t \geq 0 \} \) be a standard Brownian motion. Then \[ E\left( (W(2) + W(3))^2 \right) \] equals _______ (answer in integer).

Let \( X \) follow a 10-dimensional multivariate normal distribution with zero mean vector and identity covariance matrix. Define \( Y = \log_e \sqrt{X^T X} \) and let \( M_Y(t) \) denote the moment generating function of \( Y \) at \( t \), \( t>-10 \). Then \( M_Y(2) \) equals _________ (answer in integer).

Let \( X \sim {Bin}(2, \frac{1}{3}) \). Then \( 18 \cdot E(X^2) \) equals ________ (answer in integer).

Let \( X \) be a random variable with distribution function \( F \), such that \[ \lim_{h \to 0^-} F(3 + h) = \frac{1}{4} \quad {and} \quad F(3) = \frac{3}{4}. \] {Then } \( 16 \, {Pr}(X = 3) \) equals _________ (answer in integer).

Let \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) be a linear map defined by \[ T(x_1, x_2, x_3) = (3x_1 + 5x_2 + x_3, x_3, 2x_1 + 2x_3). \] {Then the rank of \( T \) is equal to ________ (answer in integer).}

Let \( P = \begin{pmatrix} 1 & 2 \\ -1 & 4 \end{pmatrix} \) and \( Q = P^3 - 2P^2 - 4P + 13I_2 \), where \( I_2 \) denotes the identity matrix of order 2. Then the determinant of \( Q \) is equal to ________ (answer in integer).

Let \[ I = \pi^2 \int_0^1 \int_0^1 y^2 \cos \pi(1 + xy) \, dx \, dy. \] The value of \( I \) is equal to _________ (answer in integer).

For a given data \( (x_i, y_i) \), \( i = 1, 2, \dots, n \), with \( \sum_{i=1}^{n} x_i^2>0 \), let \( \hat{\beta} \) satisfy \[ \sum_{i=1}^{n} (y_i - \hat{\beta} x_i)^2 = \inf_{\beta \in \mathbb{R}} \sum_{i=1}^{n} (y_i - \beta x_i)^2. \] {Further, let } \( v_j = y_j - x_j \) and \( u_j = 2x_j \), for \( j = 1, 2, \dots, n \), and let \( \hat{\gamma} \) satisfy} \[ \sum_{i=1}^{n} (v_i - \hat{\gamma} u_i)^2 = \inf_{\gamma \in \mathbb{R}} \sum_{i=1}^{n} (v_i - \gamma u_i)^2. \] {If } \( \hat{\beta} = 10 \), then the value of \( \hat{\gamma} \) is:

Let \( (X, Y)^T \) follow a bivariate normal distribution with
\[ E(X) = 3, \quad E(Y) = 4, \quad \text{Var}(X) = 25, \quad \text{Var}(Y) = 100, \quad \text{Cov}(X, Y) = 50 \rho, \]

Let \( (X_1, X_2, X_3) \) follow the multinomial distribution with the number of trials being 100 and the probability vector \( \left( \frac{3}{10}, \frac{1}{10}, \frac{3}{5} \right) \). 
Then \( E(X_2 | X_3 = 40) \) equals: