Probability generating function

In summary, the conversation discusses the possibility of a probability generating function (pgf) having a constant term. The individual asking the question was asked to show the derivative of a pgf was equal to a multiple of the pgf, and they wondered if adding a constant term would still satisfy the equation. The answer is yes, but it is important to note that this would no longer be a valid pgf as the sum of the coefficients should always be 1. More context or the actual question would be helpful in providing a more complete answer.
  • #1
Poirot1
245
0
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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  • #2
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
 

FAQ: Probability generating function

What is a probability generating function?

A probability generating function (PGF) is a mathematical function used to describe the probability distribution of a discrete random variable. It is a generating function that can be used to calculate the probabilities of different outcomes of a random variable.

How is a probability generating function related to a moment generating function?

A probability generating function is a special case of a moment generating function, where the random variable is discrete. The PGF can be obtained from the moment generating function by substituting the variable with z=e^t.

What is the purpose of a probability generating function?

The main purpose of a probability generating function is to provide a concise and efficient way of describing the probability distribution of a discrete random variable. It can also be used to calculate moments and other properties of the distribution.

How is a probability generating function used in practice?

In practice, a probability generating function is used to model and analyze a wide range of phenomena that can be described by discrete random variables, such as coin tosses, dice rolls, and number of customers in a queue. It is also commonly used in fields such as statistics, finance, and engineering.

Can a probability generating function be used for continuous random variables?

No, a probability generating function is only applicable for discrete random variables. For continuous random variables, a different type of generating function called the characteristic function is used.

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