How to find pdf given moment generating function

In summary, to find the pdf from a moment generating function, you need to take the inverse Fourier transform of the moment generating function. However, this method can only be used if the moment generating function exists. The moment generating function is a mathematical function used to determine the moments of a probability distribution, including the mean and variance. It can be used to find the pdf, but they are not directly related. There are some limitations to using the moment generating function, such as its existence and efficiency, as well as potential limitations on the values of the random variable.
  • #1
math-chick_41
34
0
Given that the moment generating function of a random variable is
(e^t)/(2-e^t) is there a way I can go backwards and find the pdf, or could 2 different pdf's have the same mgf?
 
Physics news on Phys.org
  • #2
Any bonafide moment generating function always maps to a unique distribution.
The random variable for the moment generating functi`on that you have given takes values at 1,2,3...infinity with probability 1/2,1/4,1/8...ie p(k)=1/2^k for k in natural numbers and zero otherwise.
 
  • #3


The moment generating function (MGF) is a useful tool in probability theory that allows us to find the moments of a random variable. However, it is not always possible to go backwards and find the probability density function (PDF) from the MGF. In fact, there can be multiple PDFs that have the same MGF. This is known as the "moment problem" and it arises because the MGF only contains information about the moments of a random variable, not the entire distribution.

To find the PDF from the MGF, we can use the inverse Fourier transform or the inverse Laplace transform. However, this may not always lead to a unique solution. In some cases, it may be impossible to find the PDF from the MGF, especially if the MGF does not exist for certain values of t.

In the given example, the MGF is (e^t)/(2-e^t). This MGF is well-defined for all values of t, so we can try to find the PDF using the inverse Laplace transform. However, this may not lead to a unique solution. There could be multiple PDFs that have the same MGF, such as a uniform distribution or a mixture of distributions.

So, in conclusion, it is not always possible to find the PDF from the MGF. Even if we can find the PDF, it may not be a unique solution. Therefore, it is important to keep in mind that the MGF only contains information about the moments of a random variable and not the entire distribution.
 

1. How do I find the pdf from a moment generating function?

To find the pdf from a moment generating function, you need to take the inverse Fourier transform of the moment generating function. This will give you the characteristic function, which can then be used to calculate the pdf using the inverse Fourier transform.

2. Can a moment generating function always be used to find the pdf?

No, a moment generating function can only be used to find the pdf if it exists. If the moment generating function does not exist, then the pdf cannot be determined using this method.

3. How does the moment generating function relate to the pdf?

The moment generating function is a mathematical function that is used to determine the moments of a probability distribution. The pdf, on the other hand, is a function that describes the probability of a random variable taking on a specific value. The moment generating function can be used to find the pdf, but they are not directly related.

4. Can I use a moment generating function to find the mean and variance?

Yes, the moment generating function can be used to find the moments of a probability distribution, which include the mean and variance. This is because the moments are derived from the derivatives of the moment generating function.

5. Are there any limitations to using the moment generating function to find the pdf?

Yes, there are some limitations to using the moment generating function to find the pdf. It can only be used if the moment generating function exists, and it may not always be the most efficient method for finding the pdf. Additionally, the moment generating function may not be defined for all values of the random variable, which can also limit its usefulness.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
823
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
611
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
15
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
0
Views
990
  • Set Theory, Logic, Probability, Statistics
Replies
16
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
Back
Top