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Unformatted text preview: 1. The Fourier T ransform of the p robability density, P(x) is +â T (k) = âŤ ( e^(ikx)}*P(x)
â dx and is called the characteristic function of the r andom variable x. Let F(k) = log (T(k)) and show t hat a) b) c) F(0) = 0 Fâ(0) = i<x> Fââ (0) = i<( â x)^2> 2. Take P(x) to be the Gaussian distribution: P (x) = {1/[(2 Pi)^.5 * ( Ď )]}* e^{(xxo)^2/(2Ď ^2) Calculate the Characteristic function (see above) and obtain <x> and <(â x)^2> using the F(x) from part 1 above. ...
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This note was uploaded on 01/23/2011 for the course PHYS 123 taught by Professor Smith during the Spring '07 term at UC Davis.
 Spring '07
 SMITH

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