Concept Architecture
Concept
Theoretically, Poisson Distribution is a discrete probability distribution that describes the probability of observing a given number of independent events occurring within a fixed interval of time, space, or population when events occur randomly at a constant average rate. It is derived from Poisson process theory and is widely used to model count data. The distribution assumes that events occur independently, the probability of more than one event occurring simultaneously is negligible, and the expected event rate remains constant over the observation period.
Mathematically, the Poisson Distribution is characterised by a single parameter, ?, representing both the expected number of events and the variance. The probability mass function specifies the probability of observing exactly k events within the interval. Owing to its simple mathematical properties, the Poisson Distribution forms the theoretical basis for numerous statistical models, including Poisson regression and count process modelling.
In practice, the Poisson Distribution is estimated by calculating the observed event rate from empirical data. In health economics it is frequently used to model hospital admissions, emergency department attendances, adverse events, disease incidence, prescription counts, healthcare utilisation, and resource consumption. The model is appropriate when observed counts are relatively rare and approximately equidispersed, although alternative distributions such as the Negative Binomial Distribution are preferred when overdispersion is present.
Purpose
Used to model the probability and expected frequency of count data occurring within fixed observation periods, supporting statistical inference, regression modelling, health service utilisation analysis, disease incidence estimation, and economic evaluation involving event counts.
Mathematical Formulae
Primary Formula
P(X = k) = (e?? ? ??) � k!
where:
- X = random count variable
- k = observed number of events (k = 0, 1, 2, ?)
- ? = expected number of events
- e = Euler's constant (� 2.71828)
Supporting Formulae
E(X) = ?
Var(X) = ?
SD(X) = �?
Log-likelihood:
ln(L) = ? [y? ln(??) ? ?? ? ln(y?!)]
Related Mathematical Methods
- Poisson Process
- Poisson Regression
- Generalised Linear Models
- Maximum Likelihood Estimation
- Negative Binomial Distribution
- Exponential Distribution
- Goodness-of-Fit Testing
Example
A health economist analyses emergency hospital admissions for a cohort of patients. Historical data indicate an average admission rate of 2 admissions per patient per year (? = 2).
The probability that a patient experiences exactly three admissions during one year is:
P(X = 3) = (e?� ? 2?) � 3!
= (0.1353 ? 8) � 6
= 0.1804
Therefore, there is approximately an 18.0% probability that a patient experiences exactly three admissions during the year.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| POISSON.DIST | =POISSON.DIST(3,2,FALSE) | Calculate the probability of exactly three hospital admissions when the expected rate is two admissions per year. |
| POISSON.DIST | =POISSON.DIST(3,2,TRUE) | Calculate the cumulative probability of observing three or fewer events. |
| AVERAGE | =AVERAGE(B2:B501) | Estimate ? from observed healthcare event counts. |
| COUNT | =COUNT(B2:B501) | Determine the sample size for estimating event rates. |
| SUM | =SUM(B2:B501) | Calculate the total number of observed healthcare events. |
VBA (Optional)
Automate estimation of Poisson probabilities, cumulative event probabilities, and summary reports for healthcare utilisation and disease incidence analyses.
Sources
- Poisson S-D. Recherches sur la Probabilit� des Jugements en Mati�re Criminelle et en Mati�re Civile. 1837.
- Cameron AC, Trivedi PK. Regression Analysis of Count Data.
- Hilbe JM. Modeling Count Data.
- McCullagh P, Nelder JA. Generalized Linear Models.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Drummond MF, et al. Methods for the Economic Evaluation of Health Care Programmes.
- NICE Health Technology Evaluation Manual.
- ISPOR Good Practice Reports.
Related Concepts (2)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is the Poisson distribution?
A discrete probability distribution describing the number of times an event occurs within a fixed interval, assuming independence and a constant average rate.
Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.
What does the Poisson distribution describe about rare events?
The Poisson distribution describes how many times an event occurs within a fixed interval of time or space when events happen independently at a constant average rate. It gives the probability of seeing any given count, such as the number of admissions in a day, and it is characterised by a single parameter that is both its mean and its variance. It suits rare, independent events, but becomes inappropriate when events cluster or vary more than this equality of mean and variance allows. Counting independent events at a steady rate is what it captures. Kirkwood and Sterne (2003) describe this distribution.
Source: Kirkwood & Sterne 2003
What are the assumptions of the Poisson distribution?
The Poisson distribution assumes that events occur independently of one another, at a constant average rate over the interval, and that two events do not occur at exactly the same instant; a key property is that its variance equals its mean. If these hold, the count of events follows a Poisson distribution. So the Poisson distribution rests on independence, a constant rate, and the equality of mean and variance, and its applicability depends on these, which is why departures such as overdispersion, where the variance exceeds the mean, or clustering of events, indicate that the Poisson may be inadequate and an alternative such as the negative binomial is needed.
Source: Casella & Berger 2002
How is the Poisson distribution used?
The Poisson distribution is used to model and analyse counts of events, such as the number of cases, admissions, or occurrences in a period, to calculate probabilities of particular counts, and to underpin Poisson regression for relating rates to predictors. So the Poisson distribution is used wherever counts of independent events at a constant rate arise, providing the basis for inference about rates and for modelling count outcomes, which makes it a foundational distribution in epidemiology and health research, though its assumption that the variance equals the mean must be checked, since overdispersed counts require the negative binomial instead.
Source: Casella & Berger 2002
What are the properties of the Poisson distribution?
The Poisson distribution is a discrete distribution for non-negative integer counts, characterised by a single parameter, its mean rate, which equals its variance; it is right-skewed for small means and approaches a normal shape as the mean grows. So the Poisson distribution's properties include a single rate parameter, the equality of its mean and variance, and its approach to normality for large means, and these properties are why it is used for count data at a constant rate and why the equality of mean and variance is a diagnostic, since overdispersion, where the variance exceeds the mean, signals that the Poisson assumption is violated.
Source: Casella & Berger 2002
When is the Poisson distribution inappropriate?
The Poisson distribution is inappropriate when its assumptions fail, particularly when the counts are overdispersed, with variance exceeding the mean, often due to unobserved heterogeneity or clustering of events, or when events are not independent or the rate is not constant. So the Poisson distribution is inappropriate for overdispersed or correlated count data, since forcing the equal-mean-and-variance assumption then understates the variability and gives invalid inference, which is why overdispersion is checked for and, when present, the negative binomial distribution, which allows the variance to exceed the mean, is used instead of the Poisson.
Source: Hilbe 2011
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 23 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-157
Stable URI · Machine-readable · Resolvable · CC BY 4.0