Law of Total Expectation

A Law of Total Expectation is a theorem that states [math]\displaystyle{ \operatorname{E} (X) = \operatorname{E} (\operatorname{E} (X \mid Y)) }[/math].

• AKA: Law of Iterated Expectations, Tower Law, Adam's Law, Smoothing Theorem, Total Expectation Theorem, Partition Theorem.
  • …
• Counter-Example(s):
  • Law of Total Probability.
• See: Sample Space, Probability Theory, Random Variable, Probability Space, Expected Value, Conditional Expected Value, Countable Set, Partition of a Set.

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References

2018a

• (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Law_of_total_expectation Retrieved:2018-9-9.
  • The proposition in probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, , Adam's law, and the smoothing theorem, among other names, states that if [math]\displaystyle{ X }[/math] is a random variable whose expected value [math]\displaystyle{ \operatorname{E}(X) }[/math] is defined, and [math]\displaystyle{ Y }[/math] is any random variable on the same probability space, then : [math]\displaystyle{ \operatorname{E} (X) = \operatorname{E} (\operatorname{E} (X \mid Y)), }[/math] i.e., the expected value of the conditional expected value of [math]\displaystyle{ X }[/math] given [math]\displaystyle{ Y }[/math] is the same as the expected value of [math]\displaystyle{ X }[/math] .

    One special case states that if [math]\displaystyle{ {\left\{A_i\right\}}_i }[/math] is a finite or countable partition of the sample space, then : [math]\displaystyle{ \operatorname{E} (X) = \sum_i{\operatorname{E}(X \mid A_i) \operatorname{P}(A_i)}. }[/math]

2018b

• (Proof Wiki, 2018) ⇒ https://proofwiki.org/wiki/Total_Expectation_Theorem Retrieved:2018-9-9.
  • QUOTE: Let [math]\displaystyle{ \mathcal E = \left({\Omega, \Sigma, \Pr}\right) }[/math] be a probability space.

    Let [math]\displaystyle{ x }[/math] be a discrete random variable on [math]\displaystyle{ \mathcal E }[/math].

    Let [math]\displaystyle{ \left\{{B_1 \mid B_2 \mid \cdots}\right\} }[/math] be a partition of [math]\displaystyle{ \omega }[/math] such that [math]\displaystyle{ \Pr \left({B_i}\right) \gt 0 }[/math] for each [math]\displaystyle{ i }[/math].

    Then:

    [math]\displaystyle{ \displaystyle E \left({X}\right) = \sum_i E \left({X \mid B_i}\right) \Pr \left({B_i}\right) }[/math]

    whenever this sum converges absolutely.

    In the above:

    [math]\displaystyle{ E \left({X}\right) }[/math] denotes the expectation of [math]\displaystyle{ X }[/math]

    [math]\displaystyle{ E \left({X \mid B_i}\right) }[/math] denotes the conditional expectation of [math]\displaystyle{ X }[/math] given [math]\displaystyle{ B_i }[/math].

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