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I have a complex stochastic model, and I want to (among other things) determine whether the output is over-dispersed, compared to what we would expect under a simpler model. Which means that it would be really nice to have a clear and well-defined "simpler model" that corresponds to the full model, in some meaningful sense. To that end, I dropped a bunch of features and, when that still left me with a complicated mess, took the limiting case as $N \rightarrow \infty$, and got this, for a given $p \in (0,\frac{1}{2})$: $$ P(X = k) = C_k p^k(1-p)^{k+1} $$ or, equivalently, for $q = p (1 - p) \in (0, \frac{1}{4})$ $$ P(X = k) = \frac{1+\sqrt{1 - 4 q}}{2}C_k q^k $$ This really looks like it ought to be a "standard" distribution of some sort - it has a simple form, a reasonably common mathematical sequence for coefficients, etc. It also has a fairly straightforward conceptual interpretation: It's the probability distribution of the total number of wins prior to gambler's ruin, for an constant-size, even-money bet with win probability $p$, starting a bankroll of 1 betting unit.

So . . . is it? A bit of searching didn't turn anything up, but I may not be looking in the right places. A characterization as a special or limiting case of a more general distribution would also be satisfactory, if that more general distribution is reasonably tractable (and might even be more useful in the long run, depending on whether that greater generality lines up with any of the features I trimmed getting to this point).

Chris Henry
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  • I am not aware of a name: it is more the kind of thing you might find in a textbook exercise, such as finding the expectation is $\frac{p}{1-2p}$ and variance $\frac{p(1-p)}{(1-2p)^3}$ when $p<\frac12$, or that $P(X<\infty)=\frac1p-1$ when $p > \frac12$. – Henry Aug 19 '22 at 09:00
  • What's the support of this distribution (the one for X)? – Glen_b Aug 19 '22 at 09:23
  • @Glen_b Sorry, should have been more explicit: The support is non-negative integers (plus $\infty$ if $p > \frac{1}{2}$, I suppose). – Chris Henry Aug 19 '22 at 20:20
  • Thanks. It would be a good idea to edit to make the support explicit in the question itself. I don't recall having seen such a distribution before, named or otherwise. If I was looking for something on it, my first thoughts would be to check the relevant Johnson & Kotz books or Feller, or Kendall & Stuart, because they all hit some of the less "standard" fare, distribution wise. – Glen_b Aug 20 '22 at 02:12
  • It occurs to me that it may be possible to show that this is a particular member of the class of power series distributions (sometimes called "PSD"), for which there are a number of handy results. – Glen_b Aug 20 '22 at 02:37

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