Probability generating function

Click For Summary
SUMMARY

A probability generating function (pgf) can indeed have a constant term, but adding a constant alters its fundamental properties. Specifically, the sum of the coefficients in a pgf must equal 1, and introducing a constant term violates this condition. The discussion emphasizes the importance of adhering to the definition of a pgf when considering its derivatives and transformations.

PREREQUISITES
  • Understanding of probability generating functions (pgfs)
  • Knowledge of calculus, specifically differentiation
  • Familiarity with the properties of polynomial functions
  • Basic concepts of probability theory
NEXT STEPS
  • Research the formal definition and properties of probability generating functions
  • Study the implications of adding constants to polynomial functions
  • Learn about the derivatives of generating functions and their applications
  • Explore examples of pgfs in probability theory and their uses in modeling
USEFUL FOR

Mathematicians, statisticians, and students studying probability theory who seek a deeper understanding of probability generating functions and their properties.

Poirot1
Messages
243
Reaction score
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.
 
Physics news on Phys.org
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
 

Similar threads

Graduate Deriving a Probability Generating Function for Independent Poisson Variables
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Understanding extinction probability in simple GWP
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Probability Generating Functions
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad On the expected value of a sum of a random number of r.v.s.
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Does the generalized gamma distribution have a mgf?
  • · Replies 1 ·
Replies
1
Views
2K
MHB Probability of misprints on each page
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Proving a multivariate normal distribution by the moment generating function
  • · Replies 5 ·
Replies
5
Views
3K
MHB Moment-Generating Function question
  • · Replies 1 ·
Replies
1
Views
1K
A game of roulette and generating functions
  • · Replies 3 ·
Replies
3
Views
1K
High School Distribution of particles, given a function
  • · Replies 2 ·
Replies
2
Views
1K