Let \( X_1, X_2 \) be a random sample from \( N(\theta, 1) \) distribution, where \( \theta \in \mathbb{R} \). Consider testing \( H_0: \theta = 0 \) against \( H_1: \theta \neq 0 \). Let \( \phi(X_1, X_2) \) be the likelihood ratio test of size 0.05 for testing \( H_0 \) against \( H_1 \). Then which one of the following options is correct?
Show Hint
- \( \phi(X_1, X_2) \) is a uniformly most powerful test of size 0.05
- \( E_{\theta}(\phi(X_1, X_2)) \geq 0.05 \, \forall \, \theta \in \mathbb{R} \)
- There exists a uniformly most powerful test of size 0.05
- \( E_{\theta = 0}(X_1 \phi(X_1, X_2)) = 0.05 \)
The Correct Option is B
Solution and Explanation
Top Questions on Normal distribution
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).
- GATE ST - 2025
- Statistics
- Normal distribution
- Let \( (X_1, X_2, X_3)^T \) have the following distribution
\[ N_3 \left( \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 & 0.4 & 0 \\ 0.4 & 1 & 0.6 \\ 0 & 0.6 & 1 \end{pmatrix} \right). \]
Then the value of the partial correlation coefficient between \( X_1 \) and \( X_2 \) given \( X_3 \) is __________ (rounded off to two decimal places).- GATE ST - 2025
- Statistics
- Normal distribution
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).
- GATE ST - 2025
- Statistics
- Normal distribution
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?
- GATE ST - 2025
- Statistics
- Normal distribution
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?- GATE ST - 2025
- Statistics
- Normal distribution
Questions Asked in GATE ST exam
- Let \( X \sim \text{Bin}(3, \theta) \), where \( \theta \in (0,1) \) is an unknown parameter. For testing
\[ H_0: \frac{1}{4} \leq \theta \leq \frac{3}{4} \quad \text{against} \quad H_1: \theta < \frac{1}{4} \quad \text{or} \quad \theta > \frac{3}{4}, \]
consider the test \[ \phi(x) = \begin{cases} 1 & \text{if } x \in \{0, 3\}, \\ 0 & \text{if } x \in \{1, 2\}. \end{cases} \]
The size of the test \( \phi \) is ___________ (rounded off to two decimal places).- GATE ST - 2025
- Binomial distribution
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).- GATE ST - 2025
- Hypothesis testing
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).
- GATE ST - 2025
- Estimation
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).
- GATE ST - 2025
- Exponential distribution
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).
- GATE ST - 2025
- Generating Functions