### 6.1.4 Expected Values

You can use the probabilities to give the expected value of any numerical random variable (i.e., one whose domain is a subset of the reals). A variable's expected value is the variable's weighted average value, where its value in each possible world is weighted by the measure of the possible world.

Suppose *V* is a random variable whose domain is numerical, and *ω* is a possible
world. Define *V(ω)* to be the value *v* in the domain
of *V* such that *ω V=v*. That is, we are treating a
random variable as a function on worlds.

The **expected value** of numerical variable
*V*, written * E(V)*, is

E(V)= ∑ _{ω∈Ω}V(ω)×µ(ω)

when finitely many worlds exist. When infinitely many worlds exist, we must integrate.

**Example 6.7:**If

*number_of_broken_switches*is an integer-valued random variable,

E(number_of_broken_switches)

would give the expected number of broken switches. If the world acted according to the probability model, this would give the long-run average number of broken switches.

In a manner analogous to the semantic definition of conditional
probability, the **conditional
expected value** of variable *X* conditioned
on evidence *e*, written * E(V|e)*, is

E(V|e)= ∑ _{ω∈Ω}V(ω)×µ_{e}(ω).

Thus,

E(V|e)= (1)/(P(e))∑ _{ω e}V(ω)×P(ω)= ∑ _{ω∈Ω}V(ω)×P(ω|e ).

**Example 6.8:**The expected number of broken switches given that light

*l*is not lit is given by

_{1}E(number_of_broken_switches|¬lit(l_{1})).

This is obtained by averaging the number of broken switches over all of
the worlds in which light *l _{1}* is not lit.