Sum of Expectations of Independent Trials

Theorem
Let $$\mathcal E_1, \mathcal E_2, \ldots, \mathcal E_n$$ be a sequence of experiments whose outcomes are independent of each other.

Let $$X_1, X_2, \ldots, X_n$$ be discrete random variables on $$\mathcal E_1, \mathcal E_2, \ldots, \mathcal E_n$$ respectively.

Let $$E \left({X_j}\right)$$ be the expectation of $$X_j$$ for $$j \in \left\{{1, 2, \ldots, n}\right\}$$.

Then:
 * $$E \left({\sum_{j=1}^n X_j}\right) = \sum_{j=1}^n E \left({X_j}\right)$$

That is, the sum of the expectations equals the expectation of the sum.