User:Caliburn/s/prob/Expectation of Linear Transformation of Random Variable/General Case

Theorem

Let $X$ be a random variable.

Let $a, b$ be real numbers.

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

Then we have:

$\expect {a X + b} = a \expect X + b$

if that expectation exists.

Proof

From Linear Transformation of Real-Valued Random Variable is Random Variable:

$a X + b$ is a real-valued random variable.

We then have:

\(\ds \expect {a X + b}\)

\(=\)

\(\ds \int \paren {a X + b} \rd \Pr\)

\(\ds \)

\(=\)

\(\ds \int a X \rd \Pr + \int b \rd \Pr\)

Integral of Integrable Function is Additive

\(\ds \)

\(=\)

\(\ds a \int X \rd \Pr + b \int 1 \rd \Pr\)

Integral of Integrable Function is Homogeneous

\(\ds \)

\(=\)

\(\ds a \expect X + b \map \Pr \Omega\)

Definition of Expectation: General Case, Integral of Characteristic Function: Corollary

\(\ds \)

\(=\)

\(\ds a \expect X + b\)

Definition of Probability Space