Probability generating functions
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Binomial Distribution
Recall that the discrete random variable X follows the Binomial distribution when it arises from the application of n successive trials to a situation where at each trial there is a probability p of a success and q = 1-p of a failure. Then the probability of r successes is given by [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] We write [Equation goes here &- download the original to see it.] Then the probability generating function for X is [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] comes from expanding the expression [Equation goes here &- download the original to see it. by the Binomial Theorem. Hence, the probability generating function for [Equation goes here &- download the original to see it.] is [Equation goes here &- download the original to see it.] Thus, the Binomial distribution is that distribution whose probability generating function is [Equation goes here &- download the original to see it.] We now use the properties of a probability generating function to find E(X) and Var(X) for [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.] [Equation goes here &- download the original to see it.]
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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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