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 b is an 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, b is an m×n matrix and a is an m-dimensional vector. If X has mean vector μX and covariance matrix ΣX, then Y has mean vector and covariance matrix
[3.30]
[3.31]
Exercises
Suppose X is a three-dimensional random vector with the parameters shown in Exhibit 3.5. Let Y = 10 + X1 + 3X2 – 2X3. 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 = A2 – 2A – 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, 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?
Solution