Question

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).

Hint:

Partial correlation coefficients measure the strength of a relationship between two variables after removing the influence of a third variable. Use the covariance matrix to compute them.

Answer 1

Step 1: Recall the formula for the partial correlation coefficient.
The partial correlation coefficient between two variables \( X_1 \) and \( X_2 \), given a third variable \( X_3 \), is given by: \[ \rho_{X_1, X_2 | X_3} = \frac{{Cov}(X_1, X_2) - {Cov}(X_1, X_3) {Cov}(X_2, X_3)}{\sqrt{({Var}(X_1) - {Cov}(X_1, X_3)^2)({Var}(X_2) - {Cov}(X_2, X_3)^2)}}. \] Step 2: Identify the covariance matrix.
From the given covariance matrix, we have: \[ {Cov}(X_1, X_2) = 0.4, \quad {Cov}(X_1, X_3) = 0, \quad {Cov}(X_2, X_3) = 0.6. \] The variances are: \[ {Var}(X_1) = 1, \quad {Var}(X_2) = 1, \quad {Var}(X_3) = 1. \] Step 3: Compute the partial correlation.
Substituting the values into the formula for partial correlation: \[ \rho_{X_1, X_2 | X_3} = \frac{0.4 - 0 \times 0.6}{\sqrt{(1 - 0^2)(1 - 0.6^2)}} = \frac{0.4}{\sqrt{(1)(1 - 0.36)}} = \frac{0.4}{\sqrt{0.64}} = \frac{0.4}{0.8} = 0.5. \] Thus, the partial correlation coefficient is \( \boxed{0.50} \).