Binomial and Poisson Distributions

The binomial and Poisson distributions are the two most used discrete distributions in biostatistics. The binomial describes the number of successes in a fixed number of independent yes/no trials, while the Poisson describes the number of events occurring in a fixed interval of time or space when events happen at a constant average rate. Both model counts, which are pervasive in health data.

Definition

The binomial distribution gives the probability of obtaining a given number of successes in a fixed number n of independent trials each with success probability p; the Poisson distribution gives the probability of a given number of events in a fixed interval when events occur independently at a constant mean rate.

Scope

The entry covers the assumptions, parameters, mean, and variance of the binomial and Poisson distributions, the settings each describes, the relationship between them, and their normal approximations. It illustrates their use for proportions and event rates in health research. It is a methodological reference and not clinical guidance.

Core questions

• What assumptions define a binomial situation versus a Poisson situation?
• How are the mean and variance of each distribution determined?
• When does the Poisson distribution approximate the binomial?
• When can each be approximated by the normal distribution?

Key concepts

• Bernoulli trial
• Number of trials n and success probability p
• Binomial mean and variance
• Poisson rate parameter
• Equality of Poisson mean and variance
• Poisson approximation to the binomial
• Normal approximation
• Counts, proportions, and event rates

Mechanisms

A binomial distribution arises from a fixed number n of independent trials, each a Bernoulli trial with the same probability p of success; the count of successes has mean np and variance np(1-p). The Poisson distribution arises as the limit of the binomial when n is large and p small while their product (the expected count) stays moderate, so it models rare events over many opportunities; it has a single parameter equal to both its mean and its variance, reflecting events occurring at a constant rate. When n is large, or when the Poisson mean is large, both distributions can be approximated by a normal distribution, which is why methods for proportions and rates often borrow normal-based confidence intervals and tests. In health research the binomial underlies the analysis of proportions, such as the number of patients responding to a treatment, while the Poisson underlies counts and incidence rates, such as the number of new cases in a population over a period.

Clinical relevance

Binomial and Poisson models underpin the analysis of proportions and event rates reported throughout the health literature, so recognising which applies aids critical reading of results on response rates and disease incidence. This entry is methodological and does not direct individual care.

Epidemiology

The Poisson distribution is the natural model for counts of relatively rare events accumulating over person-time, and so it is fundamental to the analysis of incidence rates in epidemiology; the binomial underlies the analysis of risks and proportions, such as cumulative incidence in a closed group.

History

The binomial distribution was studied by Jacob Bernoulli in his analysis of repeated trials published in 1713, and de Moivre later derived its normal approximation. Siméon Denis Poisson introduced the distribution bearing his name in 1837 as a limit of the binomial for rare events. Both became standard tools for modelling counts when statistics was applied to medicine and public health.

Key figures

• Jacob Bernoulli
• Siméon Denis Poisson
• Abraham de Moivre

Seminal works

• rosner-2015
• armitage-2002
• ross-2014

Frequently asked questions

How do I know whether to use a binomial or a Poisson model?

Use the binomial when there is a fixed number of independent yes/no trials and you count successes; use the Poisson when you count events occurring over a continuous interval of time or space at a roughly constant rate, with no fixed number of trials.

Why is the Poisson distribution's mean equal to its variance?

It follows from the distribution's structure as a limit of the binomial for rare events; this equality is also a practical check, since count data whose variance greatly exceeds its mean (overdispersion) may not fit a simple Poisson model.