Question

Let \( (Y, X_1, X_2) \) be a random vector with mean vector

\[ \begin{pmatrix} 5 \\ 2 \\ 0 \end{pmatrix} \]

and covariance matrix

\[ \begin{pmatrix} 10 & 0.5 & -0.5 \\ 0.5 & 7 & 1.5 \\ -0.5 & 1.5 & 2 \end{pmatrix} \]

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).

Hint:

For calculating the multiple correlation coefficient, use the formula involving variances and covariances from the covariance matrix.

Answer 1

Correct Answer: 0.14

The multiple correlation coefficient \( R_{Y \mid X_1, X_2} \) is calculated using the covariance matrix and the formula for the coefficient:

\[ R_{Y \mid X_1, X_2} = \sqrt{ \frac{ \text{Cov}(Y,X_1)^2 \, \text{Var}(X_2) - 2\,\text{Cov}(Y,X_1)\text{Cov}(Y,X_2)\text{Cov}(X_1,X_2) + \text{Cov}(Y,X_2)^2 \, \text{Var}(X_1) }{ \text{Var}(Y)\left(\text{Var}(X_1)\text{Var}(X_2) - \text{Cov}(X_1,X_2)^2\right) } } \]

Using the covariance matrix and calculating the necessary values gives \( R \approx 0.16 \). Thus, the value of the multiple correlation coefficient is \( 0.16 \).