Category:Definitions/Characteristic Functions of Random Variables

This category contains definitions related to Characteristic Functions of Random Variables.
Related results can be found in Category:Characteristic Functions of Random Variables.

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:

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

where:

$i$ is the imaginary unit

$\expect \cdot$ denotes expectation.