Probability generating functions
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The probability generating function of the sum of two independent variables
We begin our discussion of the sum of two independent variables by introducing an example. Example In a simulation of the Second World War London will be &'&'successfully&'&' blitzed when a five or a six has been obtained twice from the repeated throw of a six&-sided, fair dice. Let Y be the random variable representing the number of throws up to and including the occurrence of the second successful throw. i. Find the probability generating function Gx(t) for the geometric series that arises with parameter 1/3 ii.Hence, find the probability generating function of X.. iii.Find the mean of Y. Answer (i)The geometric series generated by parameter 1/3 is X~ Geo(1/3). Then [Equation goes here &- download the original to see it.] The probability generating function is [Equation goes here &- download the original to see it.] (ii) We begin by drawing a probability tree for Y&: [Diagram goes here &- download the original to see it.] Once a 5 or 6 has been thrown once the next 5 or 6 terminates the process, so after any 5, 6 the probability tree follows the geometric series with probability generating function Gx(t) of the first part of the question. Hence [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] The second part of this question could have been solved more efficiently without recourse to a diagram of the probability tree. The first and second throws of the dice are each independent of the other. Both follow the geometric distribution X~ Geo(1/3) with probability generating function [Equation goes here &- download the original to see it.] We require the probability distribution of a success in the first and second throws. Hence, we seek the probability generating function corresponding to the variable Y = X + X. The following general result would enable us to write this down immediately&: Result The probability generating function of the sum of the two independent random variables is the product of the probability generating functions of these two variables. That is [Equation goes here &- download the original to see it.] Hence, in the above example, since Y = X + X [Equation goes here &- download the original to see it.] We will now seek to justify this result that [Equation goes here &- download the original to see it.] Firstly, we ask, what is the probability distribution of the variable X +Y? It is the distribution of probabilities of each pair of events X= x i, Y=yi. That is, P(xi, yi) is the probability that X takes the value xi and Y takes the value yi. If xi and yi are independent [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.] The following diagram illustrates this process [Diagram goes here &- download the original to see it.] Following the arrows ensures that all the terms are visited once. This is an example of a diagonalisation process. The entries in the table are the entries for the distribution X+Y; they arise from the product of Gx(t) and Gy(t); hence the table demonstrates that, provided X and Y are independent [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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