List of top Statistics Questions on Bivariate 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).
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).
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 \( X \) 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 \( x_1 = 0, x_2 = 1, x_3 = 1, x_4 = 1, x_5 = 0 \) 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 \( (X, Y)^T \) 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 \( (X_1, X_2, X_3)^T \) 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).

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).

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?

Consider a Markov chain \( \{ X_n : n = 1, 2, \dots \} \) with state space \( S = \{1, 2, 3\} \) and transition probability matrix
\[ P = \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2} <br> \frac{1}{3} & 0 & \frac{2}{3} <br> \frac{2}{5} & \frac{3}{5} & 0 \end{pmatrix}. \]
Define
\[ \pi = \left( \frac{18}{67}, \frac{24}{67}, \frac{25}{67} \right). \]
Which of the following options 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?

Let \( X_1, X_2, \dots, X_5 \) be i.i.d. random vectors following the bivariate normal distribution with zero mean vector and identity covariance matrix. Define the \( 5 \times 2 \) matrix \( X = (X_1, X_2, \dots, X_5)^T \). Further, let \( W = (W_{ij}) = X^T X \), and \[ Z = W_{11} + 4W_{12} + 4W_{22}. \] Then \( {Var}(Z) \) equals:
Let \( X \) be a continuous random variable with probability density function \[ f(x) = \frac{1}{\sigma x \sqrt{2\pi}} \exp\left( -\frac{1}{2} \left( \frac{\log x - \mu}{\sigma} \right)^2 \right), \quad x>0, \] where \( \mu \in \mathbb{R}, \sigma>0 \). If \( \log_e \left( \frac{E(X^2)}{(E(X))^2} \right) = 4 \), then \( {Var}(\log_e X) \) equals:

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 \( X \) 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 \( \{ X_n \}_{n \geq 1} \) be a sequence of i.i.d. random variables with common distribution function \( F \), and let \( F_n \) be the empirical distribution function based on \( \{ X_1, X_2, \dots, X_n \} \). Then, for each fixed \( x \in (-\infty, \infty) \), which one of the following options is correct?
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?