Expectation is Linear

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

Let $$X$$ and $$Y$$ be discrete random variables on $$\left({\Omega, \Sigma, \Pr}\right)$$.

Let $$E$$ denote the expectation function.

Then:
 * $$\forall \alpha, \beta \in \R: E \left({\alpha X + \beta Y}\right) = \alpha E \left({X}\right) + \beta E \left({Y}\right)$$

Proof
Follows directly from Expectation of Function of Joint Probability Mass Distribution, thus:

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