Lesson 2: Linear Combinations of Random Variables

Overview Section

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 \(c_{1}\) through \(c_{p}\) that is multiplied bycorresponding variables \(X_{1}\) through \(X_{p}\). So, in the first term, we have \(c_{1}\) times \(X_{1}\) which is added to \(c_{2}\) times \(X_{2}\) and so on up to the variable \(X_{p}\). Mathematically this is expressed as the sum of j = 1, ... , p of the terms \(c_{j}\) times \(X_{j}\). The random variables \(X_{1}\) through \(X_{p}\) are collected into a column vector X and the coefficient \(c_{1}\) to \(c_{p}\) are collected into a column vector c. Hence, the linear combination can be expressed as \(\mathbf{c}'\mathbf{X}\).

The selection of the coefficients \(c_{1}\) through \(c_{p}\) is very much dependent on the application of interest and what kinds of scientific questions we would like to address.

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


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

  • 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.