I asked this on MathOverflow, but was redirected here.
In my research (coming from computer science), I have encountered a family of discrete probability distributions that seems to be some sort of generalization of the binomial distribution. A distribution in this family has a parameter $n \in \mathbb{N}$ and parameters $p_{0}, \ldots ,p_{n-1} \in [0,1]$. The distribution has $n+1$ possible values.
Denote the respective probabilities $q_0,\ldots,q_n$. So far, I have managed to obtain the descriptions for the first few $n$, which are the following:
- If $n=1$, the distribution is: \begin{align*}q_0 =& 1-p_0\\q_1 =& 1-q_0\end{align*}
- If $n=2$, the distribution is: \begin{align*}q_0=&(p_0-1)^2\\ q_1=&(p_0-1)p_0(p_1-1)\\ q_2 =& 1-q_0-q_1\end{align*}
- If $n=3$, the distribution is: \begin{align*}q_0=&(1 - p_0)^3\\q_1 =& 3 (p_0-1)^2 p_0 (p_1-1)^2 \\ q_2=& -3 (p_0-1) p_0 (p_1-1)^2 (2 (p_0-1) p_1-p_0) (p_2-1)\\q_3 =& 1-q_0-q_1-q_2 \end{align*}
I could give the descriptions for few higher $n$, but it becomes lots of symbols quickly.
I have noticed, that when I set $p_i = 0 \; \forall i = 1,...,n-1$ it degenerates to the distribution $\mathrm{Bin}(n,p_0)$. I have several goals with this.
First, I wanted to ask if this resembles some well known distribution family generalizing the binomial distribution? Can anyone suggest possible general formulas that would produce $q_i$ for a given $n$?
Second, assuming we have a generalization, what would be the measure of all the distributions representable as a member of this family for given $n$ or some upper bound of such measure? Denoting this measure $l_n$, I have computed that: $$\begin{aligned} l_1 &= \sqrt{2}, &l_2 &=\frac{1}{\sqrt{3}}, & l_3 &= \frac{27}{280}, & l_4 &=\frac{73162}{5010005\sqrt{5}} \end{aligned}$$.