Bernoulli distribution with non integer number of trials Can we generalise the Binomial distribution for a non-integer number of Bernoulli trials?

I see at least two (even three) ways to answer your question: 


*

*Negatively : There exists a concept of infinitely divisible distribution https://en.wikipedia.org/wiki/Infinite_divisibility_%28probability%29 ; alas,  binomial distribution do not belong to this category. 

*Positively : one can see the issue in a larger perspective :
a) either by considering the natural generalization of (discrete) Binomial Distribution B(n,p) into (continuous) Normal distribution, with pdf $\dfrac{1}{\sigma \sqrt{2 \pi}}e^{-(x-np)^2/(2 \sigma^2)}$ with $\sigma^2=npq$ and taking $n=9/2$ for example.
b) or, if one desires to stay with discrete random variables, define (artificially!!!) $X \sim Bin(4.5,p)$, by taking $X=Y/2$ with $Y \sim Bin(9,p)$.
