4.2.5 Expected Values

For a discrete random variable $X$, $E\left(X\right)$ was obtained by summing $x\cdot p\left(x\right)$ over possible $X$ values. Here we replace summation by integration and the pmf by the pdf to get a continuous weighted average.

expected value

The expected or mean value of a continuous rv $X$ with $\mathrm{pdf}f\left(x\right)$ is

$${\mu }_X=E\left(X\right)={\int }_{-\infty }^{\infty }x\cdot f\left(x\right)dx$$

EX 4.10 expected value of EX 4.9

When the pdf $f\left(x\right)$ specifies a model for the distribution of values in a numerical population, then $\mu$ is the population mean, which is the most frequently used measure of population location or center.

Often we wish to compute the expected value of some function $h\left(X\right)$ of the rv $X$ . If we think of $h\left(X\right)$ as a new rv $Y$, techniques from mathematical statistics can be used to derive the pdf of $Y$, and $E\left(Y\right)$ can then be computed from the definition. Fortunately, as in the discrete case, there is an easier way to compute $E\left[h\left(X\right)\right]$ .

expected value of function

If $X$ is a continuous $\mathrm{rv}$ with $\mathrm{pdf}f\left(x\right)$ and $h\left(X\right)$ is any function of $X$, then

$$E\left[h\left(X\right)\right]={\mu }_{h\left(X\right)}={\int }_{-\infty }^{\infty }h\left(x\right)\cdot f\left(x\right)dx$$

That is, just as $E\left(X\right)$ is a weighted average of possible $X$ values, where

• the weighting function is the pdf $f\left(x\right)$,
• $E\left[h\left(X\right)\right]$ is a weighted average of $h\left(X\right)$ values.

EX 4.11 expected value of max

In the discrete case, the variance of $X$ was defined as the expected squared deviation from $\mu$ and was calculated by summation. Here again integration replaces summation.

variance

The variance of a continuous random variable $X$ with $\mathrm{pdf}f\left(x\right)$ and mean value $\mu$ is

$$\begin{array}{rl}{\sigma }_X^2& =V\left(X\right)\\ & ={\int }_{-\infty }^{\infty }{\left(x-\mu \right)}^2\cdot f\left(x\right)dx\\ & =E\left[{\left(X-\mu \right)}^2\right]\end{array}$$

The standard deviation (SD) of $X$ is ${\sigma }_X=\sqrt{V\left(X\right)}.$

The variance and standard deviation give quantitative measures of how much spread there is in the distribution or population of $x$ values. Again $\sigma$ is roughly the size of a typical deviation from $\mu$. Computation of ${\sigma }^2$ is facilitated by using the same shortcut formula employed in the discrete case.

Proposition

$$V\left(X\right)=E\left(X^2\right)-{\left[E\left(X\right)\right]}^2$$

EX 4.12 EX 4.10 continued

Expected value and variance of linear function