3.3.2 The Expected Value of a Function

Sometimes interest will focus on the expected value of some function $h\left(X\right)$ rather than on just $E\left(X\right)$ .

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An easy way of computing the expected value of $h\left(X\right)$ is suggested by the following example.

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Proposition

• If the $\mathrm{rv}X$ has
  • a set of possible values $D$
  • $\mathrm{pmf}p\left(x\right)$ ,
• then the expected value of any function $h\left(X\right)$ ,
  • denoted by $E\left[h\left(X\right)\right]$ or ${\mu }_{h\left(X\right)}$ ,
  • is computed by $E\left[h\left(X\right)\right]=\sum _{D}h\left(x\right)\cdot p\left(x\right)$

That is,

• $E\left[h\left(X\right)\right]$ is computed in the same way that $E\left(X\right)$ itself is,
• except that $h\left(x\right)$ is substituted in place of $x$ .

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