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Theorem:
For a random variable
,
![$\displaystyle {\cal E}\{x^n\} = \left.\frac{1}{j^n}\frac{d^n}{d\omega^n}\Phi(\omega)\right\vert _{\omega=0}$](img2843.png) |
(D.47) |
where
is the characteristic function of the PDF
of
:
![$\displaystyle \Phi(\omega) \isdef {\cal E}_p\{ e^{j\omega x} \} = \int_{-\infty}^\infty p(x)e^{j\omega x}dx$](img2845.png) |
(D.48) |
(Note that
is the complex conjugate of the
Fourier transform of
.)
Proof: [201, p. 157]
Let
denote the
th moment of
, i.e.,
![$\displaystyle m_i \isdef {\cal E}_p\{x^i\} \isdef \int_{-\infty}^\infty x^i p(x)dx$](img2847.png) |
(D.49) |
Then
where the term-by-term integration is valid when all moments
are
finite.
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