Timeline for Show that for a Geometric distribution, the probability generating function is given by $\frac{ps}{1-qs}$, $q=1-p$
Current License: CC BY-SA 3.0
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| Aug 23, 2017 at 3:30 | comment | added | Joel | Additionally, it's worth noting that there is one case where $s$ is meaningful. If we have a random number of $i$ trials, each with probability of failure $s$, and the number of trials $i$ is geometrically distributed, then $\pi(s)$ is the probability of all trials failing. [but for the most part, the variable $s$ is just a useful auxiliary value rather than having a meaningful interpretation by itself]. | |
| Nov 2, 2013 at 1:56 | vote | accept | JackReacher | ||
| Nov 1, 2013 at 13:00 | comment | added | whuber♦ | +1 This is a very clear explanation. Concerning pgfs: you are correct that formally $s$ is merely a placeholder to track the probabilities. (Mathematically it generates the maximal ideal in a ring of formal power series.) The beauty of generating functions is that in many cases when we substitute numbers for $s$, the infinite sum exists, creating a function that can be analyzed using methods of Calculus. This often produces further information about the original coefficients that would have been difficult to obtain otherwise. | |
| Nov 1, 2013 at 12:30 | history | edited | Drew75 | CC BY-SA 3.0 |
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| Nov 1, 2013 at 12:23 | history | edited | Drew75 | CC BY-SA 3.0 |
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| Nov 1, 2013 at 12:00 | history | answered | Drew75 | CC BY-SA 3.0 |