I am modelling an event using a Binomial distribution $y \sim \text{Binomial}(n,p)$: $$p(y) \sim \binom{n}{y}p^y(1-p)^{n-y}$$

However, in the event I am modelling, the number of trials, $n,$ is not necessarily an integer. $n$ is also small (often between 1 and 3) and strictly greater than or equal to 0. Due to these considerations, I do not wish to approximate with a Normal distribution.

My question is as follows: is there a standard extension to the Binomial distribution that allows me to model this type of event?

And if not, what could this type of distribution look like?

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    $\begingroup$ Can you describe in a little more detail the actual problem you are considering? Though somewhat tangential, there have been a couple similar questions on this site. Here is one of them. $\endgroup$ – cardinal Nov 13 '13 at 20:16
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    $\begingroup$ This may have what you're looking for. ac.inf.elte.hu/Vol_039_2013/137_39.pdf $\endgroup$ – assumednormal Nov 13 '13 at 20:59
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    $\begingroup$ I appreciate very much the edit, though I fear that, to get an adequate answer, it will be necessary to specify precisely how your random variable is generated. Much as the binomial, and the poisson, in turn, arise from very concrete data-generation mechanisms, your random variable of interest should, too. $\endgroup$ – cardinal Nov 13 '13 at 21:05
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    $\begingroup$ Without elaborating on this, I suspect it will be difficult (or impossible) to give a complete, unambiguous answer and I would not want you to be unintentionally led astray by an answer that may have to fill in unstated assumptions in a way that (unwittingly) may not comport with your particular problem. :-) $\endgroup$ – cardinal Nov 13 '13 at 21:05
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    $\begingroup$ To attempt to clarify - you have a collection of $n_i, i = 1, \dots, N$, but you don't observe them, you only know their median? And you want to form, across all the $n_i$, a (good) approximation to the dist'n of $y$, where $y$ is the total of the goals scored by a player across all $N$ matches? $\endgroup$ – jbowman Nov 14 '13 at 0:26