I stumbled upon a variation of the Geometric distribution in which a failure may be of one of $n$ types and I wonder if this distribution has a name, what its properties are and what it has been applied to. The distribution is : $$ \mathbb{P}(X_1=x1, X_2=x2,\ldots X_n=x_n) = {x_1+x_2 +\ldots x_n \choose x_1, x_2,\ldots x_n} \pi_1^{x_1}\pi_2^{x_2}\ldots \pi_n^{x_n} \cdot (1 - p)^{x_1+x_2+\ldots x_n}\cdot p $$ where the 1st factor is the multinomial coefficient. Here $x_i=0,1,\ldots \infty$, $i=1,2,\dots n$ and $\pi_1+\pi_2+\ldots \pi_n = 1$. The interpretation is as follows: the probability of obtaining a success is $p$ and when this happens, the trial ends; $1-p$ is the probability of obtaining failure and $\pi_i$ is the probability that each failure is of the $i$th type. This distribution results after chaining a Geometric with a Multinomial distribution.
1 Answers
I'm pretty certain that what you're looking at is a special case of the negative multinomial distribution where the number of "halt" outcomes = 1. In general, negative multinomial distributions have multiple outcomes that can continue the distribution (ie "non-halting" outcomes) until we reach the expected number of "halt" outcomes (note that I'm trying to avoid using success/failure when talking about outcomes because that terminology can be ambiguous and confusing, and the more correct question is how many outcomes result in more trials). A negative multinomial distribution has a mass function of:
$$\mathbb{P}(X_0=x_0,X_1=x_1...X_n=x_n)=\frac{(\left(\sum_{i=0}^nx_i\right)-1)!}{(x_0-1)!\prod_{i=1}x_i!}\prod_{i=0}^np_i^{x_i}$$
where $x_0$ is the number of "halt" outcomes, and $p_0$ is the probability of a "halt" outcome. In this case, the distribution's coefficient can be considered a product of a negative binomial coefficient and a multinomial coefficient:
$$\frac{(\left(\sum_{i=0}^nx_i\right)-1)!}{(x_0-1)!\prod_{i=1}x_i!}=\binom{x_0+\left(\sum_{i=1}^nx_i\right)-1}{\sum_{i=1}^nx_i}\cdot\binom{\sum_{i=1}^nx_i}{x_1,x_2,x_3,...,x_n}$$
The top of the multinomial coefficient is cancelled out by one of the terms of the bottom of the negative binomial coefficient, giving us the coefficient on the left side. On a less rigorous note, as multinomial coefficients can be written as products of binomial coefficients, it feels very satisfying that negative multinomial coefficients are created by multiplying positive binomials with a negative binomial - that feels correct, arithmetically.
I've actually been doing some study on a specific form of this distribution, where the two non-halting outcomes have opposing values (in my case, 1 and -1). That answer goes through both the case for a single halt outcome, as well as when we look at multiple halting outcomes, which turns out to be much more complicated than you might expect!
