List of top Statistics Questions on Multivariate Normal Distribution asked in GATE ST

Let \((X_1,X_2,X_3)^T\) follow a trivariate normal distribution with mean vector \(\mu\) and covariance matrix \(\Sigma\) given by
\[\mu=\begin{pmatrix}1\\1\\1\end{pmatrix}\quad\text{and}\quad \Sigma=\begin{pmatrix}2&1&1\\1&2&1\\1&1&2\end{pmatrix}.\]
Then \(\text{Var}(X_1\mid X_2=1,X_3=-1)\) equals ______ (rounded off to two decimal places).

Suppose that \(X = (X_1, \ldots, X_p)^T\) follows \(N_p(\mu, \Sigma)\), where \(\mu \in \mathbb{R}^p\) and \(\Sigma\) is a positive definite matrix. Let \(A\) be a \(p \times p\) matrix such that \(A^T A = I_p\).
Let \(Y = (Y_1, \ldots, Y_p)^T = AX\). Then which of the following statements is/are correct?

Suppose that \(X = (X_1, \ldots, X_p)^T\) follows \(N_p(0, \Sigma)\), where \(\Sigma\) is a positive definite matrix. Then which of the following statements is/are correct?

Let \(Y = (Y_1, Y_2, Y_3)^T \sim N_3(0, I_3)\), where \(I_3\) denotes the identity matrix of order \(3\). Let
\[ A = \begin{pmatrix} \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{pmatrix} \quad \text{and} \quad B = I_3 - A. \] Let \(\chi_1^2\) denote chi-square distribution with \(1\) degree of freedom. Then which of the following statements is/are correct?