# 3.5 Linear Polynomials of Random Vectors

In applications, random variables are often defined as linear polynomials of random vectors. Let ** X** be a random vector with mean vector

**μ**and covariance matrix

**Σ**. Define random variable

*Y*as a linear polynomial

[3.26]

of ** X**, where

**is an**

*b**n*-dimensional row vector and

*a*∈ . The mean and variance of

*Y*are given by

[3.27]

[3.28]

Formulas [3.27] and [3.28] are general. They require no additional assumptions about ** X** whatsoever. We have already seen [3.27]. It appeared in Chapter 1 as formula [1.11], where we used in a number of examples.

Formulas [3.27] and [3.28] generalize for vector-valued polynomials. Let ** Y** be an

*m*-dimensional random vector defined as a linear polynomial

[3.29]

of an *n*-dimensional random vector ** X**. Here,

**is an**

*b**m×*

*n*matrix and

**is an**

*a**m*-dimensional vector. If

**has mean vector**

*X***μ**and covariance matrix

_{X}**Σ**, then

_{X}**has mean vector and covariance matrix**

*Y*[3.30]

[3.31]

###### Exercises

Suppose ** X** is a three-dimensional random vector with the parameters shown in Exhibit 3.5. Let

*Y*= 10 +

*X*

_{1}+ 3

*X*

_{2}– 2

*X*

_{3}. Calculate the mean and standard deviation of

*Y*using [3.27] and [3.28].

Suppose a random variable *Z* is equal to the sum of two other random variables *A* and *B* which are related by the functional relationship *B* = *A*^{2} – 2*A* – 4. Both *A* and *B* have a standard deviation of 3. Their correlation is 0.25. What is the standard deviation of *Z*?

Solution

Use [3.27] to prove that, in general,

[3.32]

Consider a three-dimensional random vector ** X**. Its first two components,

*X*

_{1}and

*X*

_{2,}are uncorrelated. They have standard deviations of 5 and 4, respectively. If

*X*

_{3}= 2

*X*

_{1}– 3

*X*

_{2}, what is the correlation between

*X*

_{1}and

*X*

_{3}?

Solution