Multiple and Partial Correlation - Multivariate Analysis - Statistics Previous Year Questions

Consider the objects {1,2,3,4} with the distance matrix

	1	2	3	4
1	0	1	11	5
2	1	0	2	3
3	11	2	0	4
4	5	3	4	0

Applying the single-linkage hierarchical procedure twice, the two clusters that result are

Let \(\underline{X} = (X_1, X_2, X_3)\) be a random vector following \(N_3(\underline{0}, \Sigma)\) distribution, where \[ \Sigma = \begin{bmatrix} 1 & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & 1 & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3} & 1 \end{bmatrix}. \] Then the partial correlation coefficient between \(X_2\) and \(X_3\), with fixed \(X_1\), equals ____________________ (correct up to two decimal places).

Let \((Y, X_1, X_2)\) be a random vector with mean vector \(\begin{pmatrix} 5 \\ 2 \\ 0 \end{pmatrix}\) and variance-covariance matrix \(\begin{bmatrix} 10 & 0.5 & -0.5 \\ 0.5 & 7 & 1.5 \\ -0.5 & 1.5 & 2 \end{bmatrix}\). Then the value of the multiple correlation coefficient between \(Y\) and its best linear predictor on \(X_1\) and \(X_2\) equals __________ (round off to 2 decimal places).

Let \(\boldsymbol{X} = \begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix}\) follow \(N_3(\boldsymbol{\mu}, \boldsymbol{\Sigma})\) with \(\boldsymbol{\mu} = \begin{pmatrix} 2 \\ -3 \\ 2 \end{pmatrix}\) and \(\boldsymbol{\Sigma} = \begin{bmatrix} 4 & -1 & 1 \\ -1 & 2 & a \\ 1 & a & 2 \end{bmatrix}\), 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 \((X_1, X_2, X_3)^T\) 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).