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 $\map \supp X$ and $\map \supp Y$ be the supports of $X$ and $Y$ respectively.

Let $f_{X, Y} : \map \supp X \times \map \supp Y \to \R$ 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:

\(\ds \expect {\alpha X + \beta Y}\)

\(=\)

\(\ds \int_{y \mathop \in \map \supp Y} \int_{x \mathop \in \map \supp X} \paren {\alpha x + \beta y} \map {f_{X, Y} } {x, y} \rd x \rd y\)

\(\ds \)

\(=\)

\(\ds \alpha \int_{y \mathop \in \map \supp Y} \int_{x \mathop \in \map \supp X} x \map {f_{X, Y} } {x, y} \rd x \rd y + \beta \int_{y \mathop \in \map \supp Y} \int_{x \mathop \in \map \supp X} y \map {f_{X, Y} } {x, y} \rd x \rd y\)

Linear Combination of Definite Integrals

\(\ds \)

\(=\)

\(\ds \alpha \int_{x \mathop \in \map \supp X} x \paren {\int_{y \mathop \in \map \supp Y} \map {f_{X , Y} } {x, y} \rd y} \rd x + \beta \int_{y \mathop \in \map \supp Y} y \paren {\int_{x \mathop \in \map \supp X} \map {f_{X, Y} } {x, y} \rd x} \rd y\)

rewriting

\(\ds \)

\(=\)

\(\ds \alpha \int_{x \mathop \in \map \supp X} x \map {f_X} x \rd x + \beta \int_{y \mathop \in \map \supp Y} y \map {f_Y} y \rd y\)

Definition of Marginal Probability Density Function

\(\ds \)

\(=\)

\(\ds \alpha \expect X + \beta \expect Y\)

Definition of Expectation of Continuous Random Variable

$\blacksquare$