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Math Help - Characteristic function of a product of random variables

  1. #1
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    Characteristic function of a product of random variables

    Hello, does anyone know how to calculate the characteristic function of the product of two independent gaussian random variables ?
    Any help welcome !
    Thanks,
    Jan
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  2. #2
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    Quote Originally Posted by NoHipHop View Post
    Hello, does anyone know how to calculate the characteristic function of the product of two independent gaussian random variables ?
    Any help welcome !
    Thanks,
    Jan
    I don't think there is a general theorem for that one (well, it is a kind of convolution, but not the usual one).

    The method is the following: using the independence, we can write

    \Phi_{XY}(t)=E[e^{itXY}]=E[E[e^{itXY}|Y]]=E[\Phi_X(tY)]

    and you know \Phi_X(t)=e^{imt-\sigma^2 t^2/2} (substitute with appropriate mean and variance), so you are reduced to computing E[e^{aY+bY^2}] (with appropriate a and b). Write it as an integral and you probably know how to compute that (complete the square,...). The computation is easy if the mean is 0 (use the fact that the density of the Gaussian integrates to 1).
    Last edited by Laurent; April 1st 2009 at 09:28 AM.
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