Characteristic function: Difference between revisions

Revision as of 23:00, 18 January 2004

Revision as of 23:05, 18 January 2004

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In other words, each cumulative probability function has one and only one characteristic function that corresponds to it.

Given a characteristic function ''f'', it is possible to reconstruct the corresponding cumulative probability function:

:<math>F_X(y) - F_X(x) = \lim_{\tau \to +\infty} \frac{1} {2\pi}

\int_{-\tau}^{+\tau} \frac{e^{-itx} - e^{-ity}} {it}\, \varphi_X(t)\, dt</math>

Characteristic function can also be used to find [[moment (mathematics)|moments]] of random variable. Provided that ''n''-th moment exists, ''f''characteristic function can be differentiated ''n'' times and

:<math>\operatorname{E}\left(X^n\right) = i^n\, \varphi_X^{(n)}(0)