How derive a CDF from MGF directly ?

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SUMMARY

The discussion focuses on deriving a cumulative distribution function (CDF) from a moment generating function (MGF). The standard method involving the inverse Laplace transform is deemed infeasible due to the complexity of the MGF. An alternative approach using integration or differentiation methods is sought but ultimately, the consensus is that the inverse Laplace transform is necessary. Additionally, the use of the Fourier transform to obtain the characteristic function is mentioned as a simpler alternative for deriving the probability density function (PDF).

PREREQUISITES
  • Understanding of moment generating functions (MGF)
  • Knowledge of inverse Laplace transforms
  • Familiarity with Fourier transforms
  • Concept of probability density functions (PDF)
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  • Study the process of performing inverse Laplace transforms
  • Learn about the relationship between moment generating functions and characteristic functions
  • Explore integration techniques for deriving probability distributions
  • Investigate the application of Fourier transforms in probability theory
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Statisticians, mathematicians, and data scientists interested in probability theory and the derivation of distribution functions from moment generating functions.

nikozm
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Hi,

i am trying to develop a CDF from a given MGF. The standard way of using the inverse Laplace transform etc.. is not feasible due to complexity of MGF.

I was woldering if there is another straighforward direction via integration or differentiation method to produce the CDF (or PDF) directly from MGF ?

Any help will be useful

Thank you in advance
 
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nikozm said:
Hi,

i am trying to develop a CDF from a given MGF. The standard way of using the inverse Laplace transform etc.. is not feasible due to complexity of MGF.

I was woldering if there is another straighforward direction via integration or differentiation method to produce the CDF (or PDF) directly from MGF ?

Any help will be useful

Thank you in advance

No. You need the inverse Laplace transform. In passing when I was introduced to this material, we used Fourier transform to get characteristic function (the moments are available here too). The main advantage is that to get PDF you just needed the inverse Fourier transform, which looked just like the Fourier transform (sign change in exponential).
 

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