Moment Generating Functions and Probability Density Functions

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Discussion Overview

The discussion centers on the relationship between moment generating functions (mgfs) and probability density functions (pdfs), exploring whether a pdf can be derived from an mgf. It includes technical explanations and mathematical reasoning related to transforms and properties of these functions.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose that moment generating functions have the property of uniqueness, while others argue that characteristic functions (C.F.s) are unique instead.
  • One participant notes that the Fourier transform of a density function can be obtained from the moments, and the inverse transform of the characteristic function will yield the density function.
  • A participant mentions that if the mgf exists in a neighborhood around 0, then the characteristic function can be expressed as mgf(i*t).
  • Another participant presents a specific Fourier transform equation involving sinh functions and seeks assistance in retrieving a probability density function from it.

Areas of Agreement / Disagreement

Participants express differing views on the uniqueness of moment generating functions, indicating that multiple competing views remain regarding their properties and relationships to characteristic functions and probability density functions.

Contextual Notes

There are unresolved mathematical steps related to the derivation of probability density functions from moment generating functions and the specifics of the Fourier transform mentioned in the discussion.

arunma
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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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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!
 
arunma said:
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.
 
Last edited:

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