Characteristic function

The characteristic function of a real-valued, continuous Random variable $X$ is a complex-valued function that completely describes its Probability distribution. It is defined as

\phi^{*}_{X}(t)=E[e^{itX}]=\int_{\Omega}e^{itx}f_{X}(x)\ dx

where $E[\cdot]$ is the expectation operator and $f_{X}(x)$ is the Probability density function of $X$ . Mathematically, this is the Fourier transform of the PDF.

Unlike moment-generating functions, which may not exist, the characteristic function is guaranteed to exist for any distribution with a probability density function. Like an MGF, it can be used to calculated the moments of a distribution by series expanding the exponential in a power series to get

i^{k}\mu^{*}_{k}=\left.\frac{ \partial ^{k}\phi^{*}_{X} }{ \partial t^{k} } \right|_{t=0}

For example, the Cauchy distribution has no MGF and requires using the characteristic function to find its (quite irregular) moments.