MHB Probability generating function

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A probability generating function (pgf) can indeed have a constant term, but adding a constant alters its fundamental properties. Specifically, including a constant term means the sum of the coefficients would no longer equal one, disqualifying it as a pgf. The discussion emphasizes the importance of adhering to the definition of pgfs when considering modifications. Additionally, the derivative of a pgf can yield the same result even if a constant is added, but this would not maintain the pgf's integrity. Thus, while technically possible, adding a constant to a pgf is not appropriate.
Poirot1
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My question is : can a pgf have a constant term?

The reason I ask is that I was asked to show the (time) derivative of a pgf was equal to some multiple of the pgf and hence show the pgf was as given. So naturally , I differentiated the given answer and showed it satisfied the equation. But surely I could stick a constant on the end and it would differentiate to the same thing.
 
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Poirot said:
My question is : can a pgf have a constant term?

The reason I ask is that I was asked to show the (time) derivative of a pgf was equal to some multiple of the pgf and hence show the pgf was as given. So naturally , I differentiated the given answer and showed it satisfied the equation. But surely I could stick a constant on the end and it would differentiate to the same thing.

The simple answer is yes.

The more complicated answer is: please post the actual question or more context.

Adding something onto a pgf will result in it no longer being a pgf, since the sum of the coefficients should be 1.

CB
 

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