Expectation is Linear/General Case

Theorem

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

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

Let $\expect X$ denote the expectation of $X$.

Then:

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

Proof

From Integral of Integrable Function is Homogeneous, we have:

$\alpha X$ and $\beta Y$ are $\Pr$-integrable.

From Integral of Integrable Function is Additive, we have:

$\alpha X + \beta Y$ is $\Pr$-integrable.

From Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable, we have:

$\alpha X + \beta Y$ is a real-valued random variable.

Then:

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

\(=\)

\(\ds \int \paren {\alpha X + \beta Y} \rd \Pr\)

Definition of Expectation

\(\ds \)

\(=\)

\(\ds \int \paren {\alpha X} \rd \Pr + \int \paren {\beta Y} \rd \Pr\)

Integral of Integrable Function is Additive

\(\ds \)

\(=\)

\(\ds \alpha \int X \rd \Pr + \beta \int Y \rd \Pr\)

Integral of Integrable Function is Homogeneous

\(\ds \)

\(=\)

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

Definition of Expectation

$\blacksquare$