List of top Multivariate Analysis Questions asked in GATE ST

Let $X=(X_1,X_2,X_3)^{T}\sim N_3(\mu,\Sigma)$ where \[ \mu=\begin{pmatrix}3\\2\\4\end{pmatrix},\qquad \Sigma=\begin{pmatrix} 4 & 2 & 1\\ 2 & 3 & 0\\ 1 & 0 & 2 \end{pmatrix}. \] Find $E[X_1\mid(X_2=4,X_3=7)]$ (in integer).

Let \( (3, 6)^T, (4, 4)^T, (5, 7)^T \) and \( (4, 7)^T \) be four independent observations from a bivariate normal distribution with the mean vector \( \mu \) and the covariance matrix \( \Sigma \). Let \( \hat{\mu} \) and \( \hat{\Sigma} \) be the maximum likelihood estimates of \( \mu \) and \( \Sigma \), respectively, based on these observations. Then \( \hat{\Sigma} \hat{\mu} \) is equal to

Let \( X = \begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix} \) follow \( N_3(\mu, \Sigma) \) with \( \mu = \begin{pmatrix} 2 \\ -3 \\ 2 \end{pmatrix} \) and \( \Sigma = \begin{pmatrix} 4 & -1 & 1 \\ -1 & 2 & a \\ 1 & a & 2 \end{pmatrix} \), where \( a \in \mathbb{R} \). Suppose that the partial correlation coefficient between \( X_2 \) and \( X_3 \), keeping \( X_1 \) fixed, is \( \frac{5}{7} \). Then \( a \) is equal to

Let \( X_1, X_2, \dots, X_{20} \) be a random sample of size 20 from \( N_6(\mu, \Sigma) \), with det(\(\Sigma\)) \(\neq 0\), and suppose both \(\mu\) and \(\Sigma\) are unknown. Let \[ \bar{X} = \frac{1}{20} \sum_{i=1}^{20} X_i \quad \text{and} \quad S = \frac{1}{19} \sum_{i=1}^{20} (X_i - \bar{X})(X_i - \bar{X})^T. \] Consider the following two statements: \begin{enumerate} \item The distribution of \( 19S \) is \( W_6(19, \Sigma) \) (Wishart distribution of order 6 with 19 degrees of freedom). \item The distribution of \( (X_3 - \mu)^T S^{-1} (X_3 - \mu) \) is \( \chi^2_6 \) (Chi-square distribution with 6 degrees of freedom). \end{enumerate} Then which of the above statements is/are true?