THER3A6-1 - 1 The Fourier T ransform of the p robability...

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Unformatted text preview: 1. The Fourier T ransform of the p robability density, P(x) is +∞ T (k) = ∫ ( e^(ikx)}*P(x) -∞ 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 F’(0) = i<x> F’’ (0) = i<( ∆ x)^2> 2. Take P(x) to be the Gaussian distribution: P (x) = {1/[(2 Pi)^.5 * ( σ )]}* e^-{(x-xo)^2/(2σ ^2) Calculate the Characteristic function (see above) and obtain <x> and <(∆ x)^2> using the F(x) from part 1 above. ...
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THER3A6-1 - 1 The Fourier T ransform of the p robability...

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