4

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?

Andrew
  • 1,150
  • 6
    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. – cardinal Nov 13 '13 at 20:16
  • 1
    This may have what you're looking for. http://ac.inf.elte.hu/Vol_039_2013/137_39.pdf – assumednormal Nov 13 '13 at 20:59
  • 3
    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. – cardinal Nov 13 '13 at 21:05
  • 3
    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. :-) – cardinal Nov 13 '13 at 21:05
  • In the actual system, there are multiple trials, the median number of goals scored in a match, and I have a point-estimate (which I'm not questioning) of the probability of success of the first trial. A success in the first trial represents that the first goal was scored by a given player. Given that I have $n \in N$ trials, it is easy to see that a Binomial random variable can be used to model whether a given player, with known probability of scoring the first goal, will score any goal at all (assuming independent and identical probabilities of scoring a goal, which is reasonable)... – Andrew Nov 13 '13 at 21:46
  • It is understood that since the number of goals is a natural number, there are multiple values that can be a median. In practice, however, I have found that there is a particular value for the median that makes more sense (depending on the match). Of course, rounding to the nearest integer is not correct. In Mathematics and Statistics, there have been several extensions to known formulas to support a larger domain and I was curious on how one could extend the Binomial to allow for $n \in R+$. – Andrew Nov 13 '13 at 21:51
  • Agree with @cardinal where you imply that there are many possible answers/extensions. It seems like there is no consensus about a single "standard" extension. I'm also interested in the same argument from a theoretical perspective (and some may even have a similar problem in another domain), so all possible extensions are welcome. – Andrew Nov 13 '13 at 22:04
  • 2
    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? – jbowman Nov 14 '13 at 00:26

0 Answers0