Expectation is Linear/Continuous

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$.

Let $E$ denote the expectation function.

Then:
 * $\forall \alpha, \beta \in \R: \expect {\alpha X + \beta Y} = \alpha \expect X + \beta \expect Y$

Proof
Let $\map {\operatorname {supp} } X$ and $\map {\operatorname {supp} } Y$ be the supports of $X$ and $Y$ respectively.

Let $f_{X, Y} : \map {\operatorname {supp} } X \times \map {\operatorname {supp} } Y \to \R$ be the joint probability density function of $X$ and $Y$.

Let $f_X$ and $f_Y$ be the marginal probability density functions of $X$ and $Y$.

Then: