Characteristic function

The characteristic function of a random variable $X$ is defined by:

$${\Phi }_X(\omega )=E\left(e^{j\omega X}\right)={\int }_{-\infty }^{\infty }{f}_X(x)e^{j\omega x}dx$$

where $E$ is the expected value, $\omega$ is an unspecified value. We can essentially view ${\Phi }_X(\omega )$ as the continuous-time Fourier transform of the pdf $f_X(x)$. Then, from the inverse CTFT, the pdf of $X$ is given by:

$$f_X(x)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\Phi }_X(\omega )e^{-j\omega x}d\omega$$

i.e., every pdf and its characteristic function form a unique Fourier transform pair. This also holds in the discrete-time case as well.

$${\Phi }_X(\omega )=\sum _{k=-\infty }^{\infty }{p}_X(k)e^{j\omega k}$$ $$p_X(k)=\frac{1}{2\pi }{\int }_0^{2\pi }{\Phi }_X(\omega )e^{-j\omega k}d\omega ,k\in \mathbb{Z}$$

so even in the DT case, we can recover the exact probabilities.

The moment theorem tells us that:

$$E(X^n)=\frac{1}{j^n}\frac{d^n}{d{\omega }^n}{\Phi }_X(\omega ){|}_{\omega =0}$$

This allows us to find the mean and variance of a given probability distribution.