m11071 - Connexions module m11071 1 Characteristic...

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Unformatted text preview: Connexions module: m11071 1 Characteristic Functions * Nick Kingsbury This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License † Abstract This module introduces characteristic functions. You have already encountered the Moment Generating Function of a pdf in the Part IB probability course. This function was closely related to the Laplace Transform of the pdf. Now we introduce the Characteristic Function for a random variable, which is closely related to the Fourier Transform of the pdf. In the same way that Fourier Transforms allow easy manipulation of signals when they are convolved with linear system impulse responses, Characteristic Functions allow easy manipulation of convolved pdfs when they represent sums of random processes. The Characteristic Function of a pdf is de ned as: Φ X ( u ) = E e iux = R ∞-∞ e iux f X ( x ) dx = F {- u } (1) where F { u } is the Fourier Transform of the pdf. Note that whenever f X is a valid pdf, Φ(0) = R f X ( x ) dx = 1 Properties of Fourier Transforms apply with- u substituted for ω . In particular: • Convolution- (sums of independent rv's) Y = N X i =1 X i ⇒ f Y = f X 1 * f X 2 * ··· * f X N ⇒ Φ Y ( u ) = N Y i =1 (Φ X i ( u )) (2) • Inversion f X ( x ) = 1 2 π Z e- ( iux ) Φ X ( u ) du (3) • Moments d n du n Φ X ( u ) = Z ( ix ) n e iux f X ( x ) dx ⇒ E [ X n ] = Z x n f X ( x ) dx = 1 i n d n du n Φ X ( u ) | u =0 (4) * Version 2.3: Jun 7, 2005 4:35 pm GMT-5 † http://creativecommons.org/licenses/by/1.0 http://cnx.org/content/m11071/2.3/ Connexions module: m11071 2 • Scaling If Y = aX , f Y ( y ) = f X ( x ) a from this equation in our previous discussion of functions of random variables, then Φ Y ( u ) = R e iuy f Y ( y ) dy = R e iuax f X ( x )...
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