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


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,

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∈ΩV(ω)×µe(ω).


E(V|e) =(1)/(P(e))∑ω  |= eV(ω)×P(ω)
=ω∈ΩV(ω)×P(ω|e ).
Example 6.8: The expected number of broken switches given that light l1 is not lit is given by

This is obtained by averaging the number of broken switches over all of the worlds in which light l1 is not lit.