Characteristic Function

Definition

Let

• $X$ : Random variable
• $i$ : Imaginary unit
• $t\in \mathbb{R}$

If $\phi (t)=E(e^{itX})$

Then we say $\phi (t)$ is the characteristic function of $X$

Remark

Important property of this expectation is that while distributions may not have an mgf, every distribution has a unique characteristic function.