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:

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

$\blacksquare$


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