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In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

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Solving recursion / finding generating function of a probability mass function

I do not think it is possible to get a closed-form of either $f_n$ or $F_n$. For brevity, let's rewrite your recurrence equation for the generating function $a_n:=F_n(x)$ as $a_n = q a_{n-1} + p a_{n …
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