Multiplication Property of Characteristic Functions

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Theorem

Let $X$ and $Y$ be independent random variables on a probability space $\left({\Omega, \Sigma, \Pr}\right)$.

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 $E$ be the expectation operator.

\(\displaystyle \phi_{X + Y} \left({t}\right)\) \(=\) \(\displaystyle E \left({e^{i t \left({X + Y}\right)} }\right)\) Definition of Characteristic Function
\(\displaystyle \) \(=\) \(\displaystyle E \left({e^{i t X} e^{i t Y} }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle E \left({e^{ i t X} }\right) E \left({e^{ i t Y} }\right)\) Functions of Independent Random Variables are Independent, Expected Value of Product is Product of Expected Value
\(\displaystyle \) \(=\) \(\displaystyle \phi_X \left({t}\right) \phi_Y \left({t}\right)\)

Hence:

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

$\blacksquare$


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