N are independent, but " not " identically distributed random variables, where G _ { X _ i } denotes the probability generating function of X _ i, then
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*The probability generating function of a binomial random variable, the number of successes in " n " trials, with probability " p " of success in each trial, is
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:It also follows that the probability generating function of the difference of two independent random variables " S " = " X " 1 & minus; " X " 2 is
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*Suppose that " N " is also an independent, discrete random variable taking values on the non-negative integers, with probability generating function " G " " N ".
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In the following W _ N ( x ) \, denotes the probability generating function of " N " : for this see the table in ( a, b, 0 ) class of distributions.
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*The probability generating function of a negative binomial random variable on { 0, 1, 2 . . . }, the number of failures until the " r " th success with probability of success in each trial " p ", is
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The work is focused on the properties of this distribution for instance a necessary condition on the parameters and their maximum likelihood estimators ( MLE ), the analysis of the probability generating function ( PGF ) and how it can be expressed in terms of the coefficients of ( modified ) Hermite polynomials.
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The probability generating function of non-negative integer-valued random variable leads to the probability generating functional being defined analogously with respect to any non-negative bounded function \ textstyle v on \ textstyle \ textbf { R } ^ d such that \ textstyle 0 \ leq v ( x ) \ leq 1.
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