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Probability density function via its characteristic help

  1. May 5, 2012 #1
    Hi there,
    This is my first post...
    and be kind on my english please...:)

    So here is a problem i cannot solve...I cant reach to something satifactory
    your ideas would be very helpful

    1. The problem statement, all variables and given/known data
    The probability density function f×(x) of the random variable X is zero when x<α, α>0.
    Its characteristic function is Φ(ω).
    Prove than if you know only the real part the characteristic Φ(ω) it is possible to find f×(x).
    Is that possible when α<0????

    2. Relevant equations
    So I know that the characteristic function is:

    Φ(ω)=∫ejωx*f×(x)dx

    3. The attempt at a solution

    So we know that ejωx= cos(ωχ)+j*sin(ωx)


    and Φ(ω)=∫[cos(ωx)+j*sin(ωx)]f×(x)dx

    and the real part of the charcteristic is Re(Φ)=∫cos(ωx)*fx(x)dx...

    and I now that fx(x) is the inverse Fourier transformation of the characteristic:
    fx(x)=(1/2π)∫e-jωχ*Φ(ω)dω

    and e-jωx=cos(ωx)-jsin(ωx)

    so:
    fx(x)=(1/2π)*∫cos(ωx)Φ(ω)dω - (1/2π)j∫sin(ωx)Φ(ω)dω
    but we need to show that we can find the pdf via the Real part of its characteristic:

    so (1/2π)∫cos(ωx)Φ(ω)dω=....

    so can I go on from here???

    Thanks in advance,
    sorry for my english again..
    this is a very nice forum
     
    Last edited: May 5, 2012
  2. jcsd
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