Linear Polynomial of a Random Vector

Explained:

linear polynomial of a random vector
 
   

A linear polynomial of a random vector is a random variable defined as a linear polynomial of some random vector. Let X be a random vector with mean vector and covariance matrix . Define random variable Y as a linear polynomial of X:

Y = bX + a [1]

Here b is an n-dimensional row vector and a is a scalar. The mean and variance of Y are given by

= b + a

 [2]

= bb'

 [3]

where a prime '  indicates transposition. These are general result for linear polynomials of random vectors. They require no additional assumptions about X whatsoever.

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Formulas [2] and [3] generalize for vector-valued linear polynomials. Let Y be an m-dimensional random vector defined as a linear polynomial

Y = bX + a [4]

of an n-dimensional random vector X. Here b is an mn matrix and a is an m-dimensional vector. If X has mean vector and covariance matrix , then Y has mean vector and covariance matrix given by

= b + a

 [5]

= bb'

 [6]

 

Exercises

Suppose X is a three-dimensional random vector with the parameters indicated below. Let Y = 10 + X1 + 3X2 – 2X3. Calculate the mean and standard deviation of Y using [2] and [3].

component

mean

standard deviation

correlations

 

 

 

X1

X2

X3

X1

–4

1.1

1.0

 

 

X2

0

0.7

0.3

1.0

 

X3

5

0.4

0.1

–0.2

1.0

Parameters for random vector X.

[solution]

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]

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