List of top Questions asked in GATE ST

Let \( \{0, 2\} \) be a realization of a random sample of size 2 from a binomial distribution with parameters 2 and \( p \), where \( p \in (0, 1) \). To test \( H_0: p = \frac{1}{2} \), the observed value of the likelihood ratio test statistic equals _________ (round off to 2 decimal places).

Consider the simple linear regression model \[ Y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \quad i = 1, 2, \dots, n \quad (n \geq 3), \] where \( \beta_0 \) and \( \beta_1 \) are unknown parameters and \( \epsilon_i \)'s are independent and identically distributed random variables with mean zero and finite variance \( \sigma^2>0 \). Suppose that \( \hat{\beta}_0 \) and \( \hat{\beta}_1 \) are the ordinary least squares estimators of \( \beta_0 \) and \( \beta_1 \), respectively. Define \( \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \), \( S_1 = \sum_{i=1}^n (x_i - \bar{x})^2 \), where \( y_i \) is the observed value of \( Y_i, i = 1, 2, \dots, n \). Then for a real constant \( c \), the variance of \( \hat{\beta}_0 + c \) is

Let \( X_1, X_2, X_3, Y_1, Y_2, Y_3, Y_4 \) be independent random vectors such that \( X_i \) follows \( N_4(0, \Sigma_1) \) distribution for \( i = 1, 2, 3 \), and \( Y_j \) follows \( N_4(0, \Sigma_2) \) distribution for \( j = 1, 2, 3, 4 \), where \( \Sigma_1 \) and \( \Sigma_2 \) are positive definite matrices. Further, let \[ Z = \Sigma_1^{-1/2} X X^T \Sigma_1^{-1/2} + \Sigma_2^{-1/2} Y Y^T \Sigma_2^{-1/2}, \] where \( X = [X_1 \, X_2 \, X_3] \) is a \( 4 \times 3 \) matrix, \( Y = [Y_1 \, Y_2 \, Y_3 \, Y_4] \) is a \( 4 \times 4 \) matrix and \( X^T \) and \( Y^T \) denote transposes of \( X \) and \( Y \), respectively. If \( W_m(n, \Sigma) \) denotes a Wishart distribution of order \( m \) with \( n \) degrees of freedom and variance-covariance matrix \( \Sigma \), then which one of the following statements is true?

Evaluate the limit: \[ \lim_{n \to \infty} \frac{(2^n + n 2^n \sin^2 \frac{n}{2})}{(2n - n \cos \frac{1}{n})} \] The value of the limit is _________ (round off to 2 decimal places).

Let \[ I = 4 \int_0^{\frac{1}{\sqrt{2}}} \int_0^x \frac{1}{\sqrt{x^2 + y^2}} \, dy \, dx \] Then the value of \( e^{l+\pi} \) is _________ (round off to 2 decimal places).

Let \( W(t) \) be a standard Brownian motion. Then the variance of \( W(1)W(2) \) equals

Let \( \{x_1, x_2, \dots, x_n\} \) be a realization of a random sample of size \( n (\geq 2) \) from a \( N(\mu, \sigma^2) \) distribution, where \( -\infty<\mu<\infty \) and \( \sigma>0 \). Which of the following statements is/are true?

Let \( X_1, X_2, \dots, X_n \) be a random sample of size \( n (\geq 2) \) from a \( N(0, \sigma^2) \) distribution. For a given \( \sigma>0 \), let \( f_\sigma \) denote the joint probability density function of \( (X_1, X_2, \dots, X_n) \) and \( S = \{ f_\sigma : \sigma>0 \} \). Let \( T_1 = \sum_{i=1}^{n} X_i^2 \) and \( T_2 = \left( \frac{1}{n} \sum_{i=1}^n X_i \right)^2 \). For any positive integer \( v \) and any \( \alpha \in (0, 1) \), let \( \chi^2_{v, \alpha} \) denote the \( (1 - \alpha) \)-th quantile of the central chi-square distribution with \( v \) degrees of freedom. Consider testing \( H_0: \sigma = 1 \) against \( H_1: \sigma>1 \) at level \( \alpha \). Then which one of the following statements is true?

Let \( X \) and \( Y \) be two random variables such that \( p_{11} + p_{10} + p_{01} + p_{00} = 1 \), where \( p_{ij} = P(X = i, Y = j) \), \( i, j = 0, 1 \). Suppose that a realization of a random sample of size 60 from the joint distribution of \( (X,Y) \) gives \( n_{11} = 10 \), \( n_{10} = 20 \), \( n_{01} = 20 \), \( n_{00} = 10 \), where \( n_{ij} \) denotes the frequency of \( (i,j) \) for \( i,j = 0,1 \). If the chi-square test of independence is used to test \[ H_0: p_{ij} = p_i p_j \text{ for } i,j = 0,1 \quad \text{against} \quad H_1: p_{ij} \neq p_i p_j \text{ for at least one pair } (i,j), \] where \( p_i = p_{i0} + p_{i1} \) and \( p_j = p_{0j} + p_{1j} \), then which one of the following statements is true?

Let \( X \) be a non-constant positive random variable such that \( E(X) = 9 \). Then which one of the following statements is true?