# Exercise Solution 3.16

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

*Z*

_{1}and

*Z*

_{2}. Each has mean 0. Like the first two components of

*, they have standard deviations of 5 and 4 respectively. Now set:*

**X**[s1]

where the μ_{i} 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

*on an equal footing. Instead of defining*

**X***X*

_{3}in terms of

*X*

_{1}and

*X*

_{2}, we define all three components as a linear polynomial of

*. Now we apply [3.31]*

**Z**[s2]

[s3]

Based upon this covariance matrix, we obtain

[s4]