Characteristic function

A characteristic function is the complex-valued function $\varphi_X:\mathbb{R}\to\mathbb{C}$ associated to a real-valued random variable $X$, defined by

It is defined using expectation and always exists (since $|e^{itX}|=1$). The characteristic function determines the distribution law of $X$, and it is closely related to the moment generating function (when the latter exists in a neighborhood of $0$).

Examples:

• If $X\sim \mathcal{N}(\mu,\sigma^2)$, then $\varphi_X(t)=\exp\!\bigl(i\mu t-\tfrac12\sigma^2 t^2\bigr)$.
• If $X\sim \mathrm{Bernoulli}(p)$, then $\varphi_X(t)=(1-p)+p\,e^{it}$.