Expectation is Linear/Discrete

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

Let $X$ and $Y$ be 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: