# Multiplication Property of Characteristic Functions

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## 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 $X$.

\(\displaystyle \map {\phi_{X + Y} } t\) | \(=\) | \(\displaystyle \expect {e^{i t \paren {X + Y} } }\) | Definition of Characteristic Function of Random Variable | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \expect {e^{i t X} e^{i t Y} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \expect {e^{i t X} } \expect {e^{ i t Y} }\) | Functions of Independent Random Variables are Independent, Expected Value of Product is Product of Expected Value | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map {\phi_X} t \map {\phi_Y} t\) |

Hence:

- $\phi_{X + Y} = \phi_X \phi_Y$

$\blacksquare$

## Sources

- 2005: Neil A. Weiss:
*A Course in Probability*: $\S 11.1$