3.3.3 Expected Value of a Linear Function

The $h\left(X\right)$ function of interest is quite frequently a linear function $aX+b$ . In this case, $E\left[h\left(X\right)\right]$ is easily computed from $E\left(X\right)$ without the need for additional summation.

Proposition

$E\left(aX+b\right)=a\cdot E\left(X\right)+b$ Or, using alternative notation, ${\mu }_{aX+b}=a\cdot {\mu }_X+b$

To paraphrase, the expected value of a linear function equals the linear function evaluated at the expected value $E\left(X\right)$ . Since $h\left(X\right)$ in EX 3.23 repurchase unsold computer is linear and $E\left(X\right)=2$ $E\left[h\left(X\right)\right]=800\left(2\right)-900=\$700$

\begin{proof}

$$\begin{array}{rl}E(aX+b)& =\sum _D(ax+b)\cdot p(x)\\ & =a\sum _{D}x\cdot p(x)+b\sum _{D}p(x)\\ & =aE(X)+b\end{array}$$

\end{proof}

Two special cases of the proposition yield two important rules of expected value.

1. For any constant $a$, $E\left(aX\right)=a\cdot E\left(X\right)$ (take $b=0$ ).
2. For any constant $b$, $E\left(X+b\right)=E\left(X\right)+b$ (take $a=1$ ).

Effects of Constants on Expected Value:

• Multiplication by a Constant:

  • When a random variable $X$ is multiplied by a constant $a$, it changes the unit of measurement (e.g., from inches to centimeters).

  • For example, if $a=2.54$:

    $$\text{New measurement}=aX$$

  • Rule 1: The expected value in the new units is given by:

    $$E[aX]=aE[X]$$

  • Addition of a Constant:

    • When a constant $b$ is added to each possible value of $X$: $$\text{New variable}=X+b$$
    • The expected value shifts by that same constant amount: $$E[X+b]=E[X]+b$$

• Implications:

  • These rules allow for straightforward adjustments of expected values when changing units or incorporating constant offsets.
  • They facilitate clearer interpretations and comparisons in various contexts, such as converting measurements or adjusting for baseline values.

• Conclusion:

  • Understanding how constants affect expected value aids in accurate calculations and meaningful interpretations in statistics and probability.