Lesson 2: Linear Combinations of Random Variables

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Introduction

This lesson is concerned with linear combinations or if you would like linear transformations of the variables. Mathematically linear combinations can be expressed as shown in the expression below:

$Y = c_1X_1 +c_2X_2 +\dots + c_pX_p = \sum_{j=1}^{p}c_jX_j = \mathbf{c}'\mathbf{X}$.

Here what we have is a set of coefficients c1 through cp that are multiplied by corresponding variables X1 through Xp. So, in the first term we have c1 times X1 which is added to c2 times X2 and so on up to the variable Xp. Mathematically this is expressed as the sum of j = 1, ... , p of the terms cj times Xj. The random variables X1 through Xp are collected into a column vector X and the coefficient c1 to cp are collected into a column vector c. Hence, the linear combination can be expressed as c′X.

The selection of the coefficients c1 through cp are very much dependent on the application of interest and what kinds of scientific questions we would like to address.

Learning objectives & outcomes

Upon completion of this lesson, you should be able to do the following:

• Interpret the meaning of a specified linear combination;
• Compute the sample mean and variance of a linear combination from the sample means, variances, and covariances of the individual variables.

Later on in this course, when we learn about multivariate data reduction techniques, interpretation of linear combinations will be of great importance.