Moment Theorem

Theorem: For a random variable $x$ ,

$${\cal E}\{x^n\} = \left.\frac{1}{j^n}\frac{d^n}{d\omega^n}\Phi(\omega)\right\vert _{\omega=0}$$ (D.47)

where $\Phi(\omega)$ is the characteristic function of the PDF $p(x)$ of $x$ :

$$\Phi(\omega) \isdef {\cal E}_p\{ e^{j\omega x} \} = \int_{-\infty}^\infty p(x)e^{j\omega x}dx$$ (D.48)

(Note that $\Phi(\omega)$ is the complex conjugate of the Fourier transform of $p(x)$ .)

Proof: [201, p. 157] Let $m_i$ denote the $i$ th moment of $x$ , i.e.,

$$m_i \isdef {\cal E}_p\{x^i\} \isdef \int_{-\infty}^\infty x^i p(x)dx$$ (D.49)

Then

$$\begin{eqnarray*} \Phi(\omega) &=& \int_{-\infty}^\infty p(x)e^{j\omega x} dx \\ &=& \int_{-\infty}^\infty p(x) \left(1 + j\omega x + \cdots + \frac{(j\omega)^n}{n!}+\cdots\right)dx\\ &=& 1 + j\omega m_1 + \frac{(j\omega)^2}{2} m_2 + \cdots + \frac{(j\omega)^n}{n!}m_n+\cdots \end{eqnarray*}$$

where the term-by-term integration is valid when all moments $m_i$ are finite.