Characteristic function

Definition

Let $X$ be a real random variable with law $\mu_X$. Let $i$ be the imaginary unit in the complex numbers. The characteristic function of $X$ is the function $\varphi_X \colon \mathbb{R} \to \mathbb{C}$ with $$\varphi_X(t) = E\left[e^{itX}\right] = E[\cos(tX)] + i \, E[\sin(tX)] = \int_{\mathbb{R}} e^{itx} \, \mu_X(dx).$$

Remarks

The expectation exists for all $t$, because $|e^{itX}| = 1$. The second form uses Euler's formula. Always $\varphi_X(0) = 1$, $|\varphi_X(t)| \le 1$, and $\varphi_X$ is uniformly continuous on $\mathbb{R}$. For $a, b \in \mathbb{R}$, $\varphi_{aX + b}(t) = e^{itb} \varphi_X(at)$. If $E[|X|^k] < \infty$, then $\varphi_X$ is $k$ times continuously differentiable and $\varphi_X^{(k)}(0) = i^k E[X^k]$.

Explanations

L1 human

The characteristic function is like the moment-generating function, but with an imaginary argument. This small change makes it finite for every distribution, so it is a tool that always works.