Probability generating functions
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Probability generating functions. Discrete random variable.
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Let X be a discrete random variable. Let [Equation goes here &- download the original to see it.] be the probability that X takes the value xi . Then the probability generating function defined for X is the function [Equation goes here &- download the original to see it.] This is a function of two variables. The first is xi the values that the random variable X can take. The second is t, which is introduced to create the idea of a family of related series. The main properties of probability generating functions are as follows. First Property [Equation goes here &- download the original to see it.] Proof [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] The last line follows since the sum of a probability distribution is 1. This first result could be said to state the obvious, but it is a property of a probability generating function, and we must include it. Second Property [Equation goes here &- download the original to see it.] Proof [Equation goes here &- download the original to see it.] Third Property [Equation goes here &- download the original to see it.] Proof [Equation goes here &- download the original to see it.] Then [Equation goes here &- download the original to see it.] Therefore [Equation goes here &- download the original to see it.] Fourth Property [Equation goes here &- download the original to see it.] Proof [Equation goes here &- download the original to see it.] We now proceed to apply these properties to the uniform (uniform), Binomial, Geometric and Poisson distributions to find the mean and variance in each case.
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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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