Probability generating functions
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Probability generating functions. Introduction.
We will introduce the idea of a probability generating function, by first considering a simple example of a discrete probability distribution. Example A discrete random variable X has the following probability distribution. [Table goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it. These are calculations that the student should have encountered at an earlier stage. We now introduce, however, a function Gx(t), called the probability generating function, which is [Equation goes here &- download the original to see it.] We can form a table for our current example as follows&: [Table goes here &- download the original to see it.] Thus [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] This illustrates that probability generating functions could be useful short&-cuts to finding E(X) and VarX. But in our current example the "short&-cut" seems far from short! The real advantage from probability generating functions comes from the ability to write certain probability distributions as powers of a generating series. We will show this later, but firstly, we offer a formal definition of a probability generating function.
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Contents of
Probability generating functions
1 Probability generating functions. Introduction.
2 Probability generating functions. Discrete random variable.
3 Probability generating functions. Uniform, discrete distribution
4 Binomial Distribution
5 Geometric Distribution
6 Poisson Distribution
7 The probability generating function of the sum of two independent variables
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