# Moment Generating Functions and Probability Density Functions

## Main Question or Discussion Point

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?

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mathman
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.

note: If the mgf exists in a neighborhood around 0 then the characteristic function = mgf(i*t)

Fourier Transforms of sinh

Hello:

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!

ssd
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?
In general, moment generating functions DO NOT have the property of uniqueness. C.F. s are unique.

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