5.5.1 The Difference Between Two Random Variables

An important special case of a linear combination results from taking $n=2,a_1=1$ , and $a_2=-1$ :

$$Y=a_1{X}_1+a_2{X}_2=X_1-X_2$$

We then have the following corollary to the proposition.

Corollary

$E\left(X_1-X_2\right)=E\left(X_1\right)-E\left(X_2\right)$ for any two rv’s $X_1$ and $X_2$ .

$V\left(X_1-X_2\right)=V\left(X_1\right)+V\left(X_2\right)$ if $X_1$ and $X_2$ are independent rv’s.

The expected value of a difference is the difference of the two expected values. However, the variance of a difference between two independent variables is the sum, not the difference, of the two variances. There is just as much variability in $X_1-X_2$ as in $X_1+X_2$ writing $X_1-X_2=X_1+\left(-1\right)X_2,\left(-1\right)X_2$ has the same amount of variability as $X_2$ itself.

EXAMPLE 5.31