Total Expectation Theorem

Theorem
Let $$\mathcal E = \left({\Omega, \Sigma, \Pr}\right)$$ be a probability space.

Let $$x$$ be a discrete random variable on $$\mathcal E$$.

Let $$\left\{{B_1 | B_2 | \cdots}\right\}$$ be a partition of $$\omega$$ such that $$\Pr \left({B_i}\right) > 0$$ for each $$i$$.

Then:
 * $$E \left({X}\right) = \sum_i E \left({X | B_i}\right) \Pr \left({B_i}\right)$$

whenever this sum converges absolutely.

In the above:
 * $$E \left({X}\right)$$ denotes the expectation of $$X$$;
 * $$E \left({X | B_i}\right)$$ denotes the conditional expectation of $$X$$ given $$B_i$$.

Proof
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