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Characteristic function

  1. Oct 5, 2006 #1
    Hi all,

    Im currently researching into Multivariate distributions, in particular Im trying to derive the characteristic function of the bivariate distribution of a gaussian. While knowing that a gaussian density function cannot be integrated how is it possible to find the characteristic function. I have been working on it but I keep bumping in to a dead end. The following is what I did in the most recent dead end: (does anybody know why the latex thing aint working? I havnt been on here for a while)

    so I have a double integral of the

    exp{i*x*t_x+i*y*t_y}*(joint density function)

    then I did the substitutions

    I then simplified it down such that I get the following in the exponential expression


    where si is the correlation coefficient, the problem is I cant complete the square because of the existance of si, would anybody know what I would need to do?

    I had looked at Wolfram Mathworld website, its very interesting how it uses Eulers formula but isnt there any other way of doing it?


  2. jcsd
  3. Oct 6, 2006 #2


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    Science Advisor
    Gold Member

    let w=au+bv and z=bu-av, where a^2+b^2=1. There will be some value of a where there will not be a cross product term for wz. Good luck!
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