Multiplication Property of Characteristic Functions

Theorem
Let $X$ and $Y$ be independent random variables on a probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\phi_X$ and $\phi_Y$ denote the characteristic functions of $X$ and $Y$ respectively.

Then:
 * $\phi_{X + Y} = \phi_X \phi_Y$

Proof
Let $i = \sqrt{-1}$.

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

Hence:
 * $\phi_{X + Y} = \phi_X \phi_Y$