< List of probability distributions

A **counting distribution**, or count distribution, is used to model random variables with a finite set of values, specifically the range of natural numbers. In practical terms, counting distributions can be used to model the probability of the occurrence of a certain event, such as the number of cars that pass through a particular intersection in a given time period. This blog post will explore what a counting distribution is, what are the most common models of counting distributions, and how they are applied in real-world scenarios.

## Counting distribution definition

Counting distributions are often used to describe the number of times a certain event occurs over a given time interval or in a given space. A perfect example of this is the Poisson distribution, which is widely used in probability theory and statistics. The Poisson distribution is a model for a count random variable that has the range ℕ_{0} (a full set of non-negative integers) and is commonly used to predict the probability of the occurrence of rare events, such as car accidents in a busy intersection, based on past data.

The probability mass function (PMF) of the Poisson distribution is:

Apart from the Poisson distribution, there are many other types of counting distributions which are equally important in the realm of probability theory. Among these is the negative binomial distribution, which is used to model the number of independent Bernoulli trials that are required to obtain a certain number of successes.

The PMF is

The geometric distribution is a special case of the negative binomial distribution with n = 1; it describes the number of independent Bernoulli trials that are required to obtain a single success (in other words, the number of failures until a certain point). The Bernoulli distribution, on the other hand, is the simplest counting distribution as it models a binary outcome, such as the flip of a coin.

One of the most flexible counting distributions is the Conway-Maxwell-Poisson (CMP) distribution [2]. The Poisson, geometric and Bernoulli distributions are all special cases the CMP, which can be described as a two-parameter generalization of the Poisson. The CMP is used to model count data that arises from a wide range of biological, physical, and engineering applications. It is particularly useful in scenarios where the mean and variance of a random variable are not equal, such as in cases where overdispersion is observed.

## Real world applications of counting distributions

Real-world applications of counting distributions are numerous, with examples ranging from manufacturing to biology. For instance, the Poisson distribution can be applied to study the number of defective products in a manufacturing facility, while the negative binomial distribution can be used to model the populations of endangered species in the wild. In addition, the geometric distribution is often used to study the occurrence of rare diseases in a population, while the Bernoulli distribution can be applied to describe the outcome of a clinical trial involving a new drug.

In conclusion, counting distributions are an important class of probability distributions that are used to model a wide range of random variables that have a finite set of values. The Poisson, geometric, Bernoulli, negative binomial, and Conway-Maxwell-Poisson distributions are some of the most commonly used counting distributions in probability theory. They are useful in a variety of real-world applications, from manufacturing to biology, and help us understand the random variables that underlie many phenomena. By studying counting distributions, we can better understand the probability of rare events and make better-informed decisions based on data.

## References

[1] Image: Ederporto, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

[2] Conway, RW, Maxwell, WL: A queuing model with state dependent service rates. J. Ind. Eng. 12, 132–136 (1962). Google Scholar.