Moment Generating Functions and Probability Density Functions

  1. I was reading that moment generating functions have the property of uniqueness. So just wondering: is there a way to get a probability density function from a moment generating function?
  2. jcsd
  3. mathman

    mathman 6,474
    Science Advisor
    Gold Member

    The Fourier transform of the density function (called the characterictic function) can be obtained from the moments. The inverse transform of the ch. func. will give you the density function back. For distribution functions without a density, it is a little more complicated.
  4. note: If the mgf exists in a neighborhood around 0 then the characteristic function = mgf(i*t)
  5. Fourier Transforms of sinh


    I am referring to 'Table of Laplace Transforms' by Roberts&Kaufman. But I cannot seem to get a soln for the following Fourier Transform to retrieve my probability density f(x)

    c2 * Integral{e^(iwx) * sinh[sqrt(2w)c1] / sinh[sqrt(2w)pi] dw} = f(x)

    where -pi< c1 <=0 and c2 is a constant that scales the integral appropriately so that f(x) is p.d.f. Thanks for your help!
  6. In general, moment generating functions DO NOT have the property of uniqueness. C.F. s are unique.
    Last edited: May 6, 2008
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