How to find pdf given moment generating function

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The moment generating function (MGF) provided, (e^t)/(2-e^t), corresponds to a unique probability distribution. This distribution is characterized by the probabilities p(k) = 1/2^k for k in the natural numbers, indicating that the random variable takes values 1, 2, 3, and so on, with probabilities 1/2, 1/4, 1/8, respectively. The discussion confirms that a bona fide MGF always maps to a unique distribution, negating the possibility of two different PDFs sharing the same MGF. Therefore, the PDF can be derived directly from the given MGF. Understanding the relationship between MGFs and their corresponding distributions is crucial in probability theory.
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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?
 
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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.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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