EXAMPLE 5.31

A certain automobile manufacturer equips a particular model with either a six-cylinder engine or a four-cylinder engine. Let $X_1$ and $X_2$ be fuel efficiencies for independently and randomly selected six-cylinder and four-cylinder cars, respectively. With ${\mu }_1=22$, ${\mu }_2=26$, ${\sigma }_1=1.2$, and ${\sigma }_2=1.5$ ,

$$E\left(X_1-X_2\right)={\mu }_1-{\mu }_2=22-26=-4$$ $$V\left(X_1-X_2\right)={\sigma }_1^2+{\sigma }_2^2={\left(1.2\right)}^2+{\left(1.5\right)}^2=3.69$$ $${\sigma }_{X_1-X_2}=\sqrt{3.69}=1.92$$

If we relabel so that $X_1$ refers to the four-cylinder car, then $E\left(X_1-X_2\right)=4$ , but the variance of the difference is still 3.69.