3.3.6 Variance of a Linear Function

The variance of $h\left(X\right)$ is the expected value of the squared difference between $h\left(X\right)$ and its expected value: V\left\lbrack {h\left( X\right) }\right\rbrack = {\sigma }_{h\left( X\right) }^{2} = \mathop{\sum }\limits_{D}\{ h\left( x\right) - E\left\lbrack {h\left( X\right) }\right\rbrack {\} }^{2} \cdot p\left( x\right) \tag{3.13}

When $h\left(X\right)=aX+b$ , a linear function, $h\left(x\right)-E\left[h\left(X\right)\right]=ax+b-\left(a\mu +b\right)=a\left(x-\mu \right)$

Substituting this into (3.13) gives a simple relationship between $V\left[h\left(X\right)\right]$ and $V\left(X\right)$

variance of linear function

$$\begin{array}{rl}V(aX+b)& ={\sigma }_{aX+b}^2=a^2\cdot {\sigma }_X^2\\ {\sigma }_{aX+b}& =|a|\cdot {\sigma }_X\end{array}$$

In particular,

$$\begin{array}{rl}{\sigma }_{aX}& =|a|\cdot {\sigma }_X,\\ {\sigma }_{X+b}& ={\sigma }_X\end{array}\text{\hspace{1em}(3.14)}$$

The absolute value is necessary because $a$ might be negative, yet a standard deviation cannot be. Usually multiplication by $a$ corresponds to a change in the unit of measurement (e.g., $\mathrm{kg}$ to $\mathrm{lb}$ or dollars to euros). According to the first relation in (3.14), the sd in the new unit is the original sd multiplied by the conversion factor. The second relation says that adding or subtracting a constant does not impact variability;

• it just rigidly shifts the distribution to the right or left.

EX 3.26 (EX 3.23 continued)