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Gaussian Distribution and Fourier Transforms

1. The Fourier Transform of the probability density, P(x) is

T(k) =  (e^(ikx)}*P(x) dx
and is called the characteristic function of the random variable x. Let F(k) = log (T(k)) and show that

a) F(0) = 0
b) F'(0) = i<x>
c) F'' (0) = i<(x)^2>

2. Take P(x) to be the Gaussian distribution:
P(x) = {1/[(2 Pi)^.5 * ()]}* e^-{(x-xo)^2/(2^2)
Calculate the Characteristic function (see above) and obtain <x> and <(x)^2> using the F(x) from part 1 above.


Solution Summary

Gaussian distribution and Fourier transforms are examined to determine the probability density.