How derive a CDF from MGF directly ?

In summary, the standard method of using the inverse Laplace transform to develop a CDF from MGF may not be feasible due to the complexity of the MGF. However, there is an alternative approach using Fourier transform to get the characteristic function and then using the inverse Fourier transform to obtain the PDF. This method may be more straightforward and easier to use.
  • #1
nikozm
54
0
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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  • #2
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).
 

1. What is a CDF and MGF?

A CDF (Cumulative Distribution Function) is a mathematical function that describes the probability of a random variable taking on a specific value or a value less than or equal to that specific value. An MGF (Moment Generating Function) is a mathematical function that provides a way to calculate the moments of a random variable.

2. How do you derive a CDF from an MGF directly?

To derive a CDF from an MGF directly, you can use the inverse Laplace transform. This involves taking the MGF and transforming it into a Laplace transform, then using the inverse Laplace transform to obtain the CDF. This method is useful for finding the CDF of complex distributions.

3. What are the advantages of deriving a CDF from an MGF directly?

Deriving a CDF from an MGF directly can provide a more efficient and accurate way of calculating the CDF, especially for complex distributions. It also allows for the calculation of higher moments of the random variable, which can be useful in certain applications.

4. Are there any limitations to deriving a CDF from an MGF directly?

One limitation is that this method may not be applicable for all types of distributions. It also requires knowledge of complex mathematical concepts such as Laplace transforms, which may be challenging for some individuals.

5. Can this method be used for both discrete and continuous distributions?

Yes, this method can be used for both discrete and continuous distributions. However, the calculations may differ slightly for each type of distribution. For discrete distributions, the Laplace transform is replaced with the Z-transform, while for continuous distributions, the Laplace transform is used.

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