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Moment Theorem


Theorem: For a random variable $ x$ ,

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

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

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

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

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Higher Order Moments Revisited