(30th-August-2020)
• 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∈ΩV(ω)×µ(ω)
• when finitely many worlds exist. When infinitely many worlds exist, we must integrate.
For example, The expected number of broken switches given that light l1 is not lit is given by
E(number_of_broken_switches|¬lit(l1)).
This is obtained by averaging the number of broken switches over all of the worlds in which light l1 is not lit.
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