Definition:Characteristic Function of Random Variable

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This page is about Characteristic Function in the context of Probability Theory. For other uses, see Characteristic Function.


Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

The characteristic function of $X$ is the mapping $\phi: \R \to \C$ defined by:

$\map \phi t = \expect {e^{i t X} }$


$i$ is the imaginary unit
$\expect \cdot$ denotes expectation.

Also see