Characteristic function

The characteristic function of a real-valued random variable $X$ is the function $\varphi_X:\mathbb R\to\mathbb C$ defined by

\varphi_X(t)=\mathbb E[e^{itX}].

Remarks

The expectation always exists because $|e^{itX}|=1$ . The characteristic function determines the distribution of $X$ . It is related to the moment-generating function when the latter exists near $0$ .

Examples

If $X\sim\mathcal N(\mu,\sigma^2)$ , then $\varphi_X(t)=\exp(i\mu t-\tfrac12\sigma^2t^2)$ .
If $X\sim\operatorname{Bernoulli}(p)$ , then $\varphi_X(t)=(1-p)+pe^{it}$ .