Probability Generating Functions. This appendix is based on Feller (, Chap. XI and Chap. XII). Let X be a random variable taking values j ∈ IN0 with P(X. Key words. probability generating functions, finite sums of independent The natural extension of the definition of a probability generating function to a. The main part of our results is derived by the method of generalized probability generating functions. In Polya's urn scheme, the system of equations is.


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A counterexample is constructed below. For nonnegative random variables which are very common probability generating functions applicationsthe domain where the moment generating function is finite is easy to understand.


Of course, there are complementary results for non-positive random variables, but such probability generating functions are much less common. Next we consider what happens to the moment generating function under some simple transformations of the random variables.

Location-scale transformations frequently arise when units are changed, such as length changed from inches to centimeters or temperature from degrees Fahrenheit to degrees Celsius.

As with the PGF, the proof for the MGF relies on probability generating functions law of exponents and the fact that the expected value of a product of independent variables is the product of the expected values: The following theorem gives the Chernoff boundsnamed for the mathematician Herman Chernoff.

These are upper bounds on the tail events of a random probability generating functions. The Characteristic Function Our last generating function is the nicest from a mathematical point of view. Once again, we assume that our random variables are real-valued.

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Many of the properties of the characteristic function are more elegant than the corresponding properties of the probability or moment generating functions, because the characteristic function always exists. This probability generating functions from the change of variables theorem probability generating functions expected value, albeit a complex version.

The Fourier transform is named for Joseph Fourierand is widely used in many areas of applied mathematics. As with other generating functions, the characteristic function completely determines the distribution. Indeed, the general inversion formula given next is a formula for computing certain combinations of probabilities from the characteristic function.

A special inversion formula holds for continuous distributions: Moreover, this can be done even when only some of the moments exist.


Next we consider how the characteristic function probability generating functions changed under some simple transformations of the variables. The proof is just like the one for the MGF: Again, the proof probability generating functions just like the one for the MGF: The final important property of characteristic functions that we will discuss relates to convergence in distribution.

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Since we probability generating functions only concerned with distributions, the random variables need not be defined on the same probability space.

There are analogous versions of the continuity theorem for probability generating functions and moment generating functions. PGFs help predict a number of properties about early outbreak behavior probability generating functions the population is still effectively infinite, including the probability of an epidemic, the size distribution after some number of generations, and the cumulative size distribution of non-epidemic outbreaks.

We show how PGFs can be used in both discrete-time and continuous-time settings, and discuss how to use these results to infer disease parameters from observed outbreaks.

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We use these to explore properties such as the final size of epidemics or even the dynamics once stochastic effects are negligible. We target this tutorial to biologists and public health probability generating functions who want to learn how to apply PGFs to invasive diseases, but it could also be used in an probability generating functions mathematics course on PGFs.

We include many exercises to help demonstrate concepts and to give practice applying the results. We summarize our main results in a few tables.