Consider a three-dimensional random vector X. Its first two components, X₁ and X₂, are uncorrelated. They have standard deviations of 5 and 4, respectively. If X₃ = 2X₁ – 3X₂, what is the correlation between X₁ and X₃?

This is an application for our multi-dimensional identity [3.31]. This may not be immediately obvious, but let’s present our problem in a slightly different manner.

Define a 2-dimensional random vector Z as having uncorrelated components Z₁ and Z₂. Each has mean 0. Like the first two components of X, they have standard deviations of 5 and 4 respectively. Now set:

where the are the unspecified means of the components of X.

We have not altered our characterization of X in any way. We have simply placed the components of X on an equal footing. Instead of defining X₃ in terms of X₁ and X₂, we define all three components as a linear polynomial of Z. Now we apply [3.31]

Based upon this covariance matrix, we obtain