What are some Poisson-like distributions over a finite range of integers? I'm writing a program in which, in any given time step, a random number of messages is sent.  The number of messages is always between $0$ and $n$.  I want to be able to control the probability, so that I can cause more or fewer messages to be most probable.
It would be natural to model this with a Poisson distribution.  We can think of the messages as generated by people who decide to talk at different moments within the time step.  By changing the mean $\lambda$ of the Poisson distribution, I'm able to move the peak of the curve to lower or higher integers.
However, a Poisson distribution assigns a probability to every integer $\ge 0$, and in my model, integers $>n$ should have zero probability.  At present, I'm using a Poisson distribution to generate random numbers of messages, with the additional rule that I return $n$ when the number returned by the Poisson function is $>n$.  This is OK when $\lambda$ is small, but when $\lambda$ is near $n$, the resulting distribution is not very Poisson-ey.
Are there any distributions and parameter ranges that you would suggest that I consider?  (Maybe some kind of beta binomial distribution?  If so, I'd appreciate suggestions about parameter ranges.)
 A: You can design your own Poisson like distribution. $P(X=k)=C(\lambda) \frac{\lambda^k}{k!}$. You just have to choose $C(\lambda)$ so that $\sum^n_{k=0} {P(k)}=1$. Wolfram alpha gives $C(\lambda)=n!\exp(-\lambda) / \Gamma(n+1,\lambda)$. The incomplete gamma function makes it a bit nasty I admit.
A: Binomial Distribution
I know the question is no longer relevant to the original poster, but I spent more time today trying to figure this out than I wanted to.
I was trying to answer a similar question, "what is the distribution of proteins with a particular degree of labeling, $k$, given the average degree of labeling, $\lambda$." My protein has 5 available sites ($N$ = 5) that can be chemically modified. If I average, say, $\lambda$ =2 modifications per protein, what percentage of my protein will have 0, 1, 2, 3, 4 or 5 modifications. Using the Poisson distribution with $\lambda > \sim 1.5$, I get a truncated distribution* that does not sum to 1.
Being a mediocre mathematician, I looked it up. In a 1972 paper, Use of Nonspecific Dye Labeling for Singlet Energy-Transfer Measurements in Complex Systems. A Simple Model, I found, "The Poisson distribution is an approximate form of the binomial distribution in the limit of small probability for an event (binding, in our case). The probability of binding maybe expressed as $\mu/M$, where $M$ is the number of possible binding sites on the protein."
"$\mu/M$" using the typical symbols would be "$\lambda/N$"
So the Poisson Distribution:
$$f(k;λ) = \frac{\lambda ^k e^{-\lambda}}{k!}$$
becomes the Binomial Distribution:
$$f(k;p,N) = \frac{N!}{k!(N-k)!}p^k(1-p)^{N-k}   ;$$
here $p = \lambda/N$ for finite $N$. The distributions look quite similar to each other, but the Binomial distributions all sum to 1.
Here are some plots of the Poisson and Binomial distributions using $\lambda$ = 0.3, 0.5, 1, 1.5, & 3.37; and $N$ = 5:
Poisson Distributions
Binomial Distributions
*I see in the comments to the original question that "truncated Poisson distribution" refers to the exclusion of zero. I am using the term here to mean a truncation at high $k$ due to finite $N$.
