Explore BrainMass
Share

Explore BrainMass

    Gaussian Distribution and Fourier Transforms

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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.

    © BrainMass Inc. brainmass.com October 9, 2019, 8:56 pm ad1c9bdddf
    https://brainmass.com/physics/mathematical-physics/gaussian-distribution-fourier-transforms-167319

    Attachments

    Solution Summary

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

    $2.19