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📊 '''Poisson distribution''' is a foundational statistical model used extensively in [[Definition:Actuarial science | actuarial science]] and insurance [[Definition:Risk modeling | risk modeling]] to describe the probability of a given number of [[Definition:Claims | claims]] or loss events occurring within a fixed period, assuming those events happen independently and at a roughly constant average rate. In the insurance context, it provides the mathematical backbone for estimating how many auto accidents, fire losses, or liability claims an [[Definition:Insurance carrier | insurer]] can expect across a [[Definition:Book of business | book of business]] during a policy year. Because insurance inherently deals with rare, discrete events spread across large populations, the Poisson distribution is one of the most natural and widely applied probability tools in the industry.

⚙️ The distribution is defined by a single parameter, lambda (λ), which represents the average number of events expected in the observation window. An [[Definition:Actuary | actuary]] estimating [[Definition:Claims frequency | claims frequency]] for a portfolio of [[Definition:Homeowners insurance | homeowners policies]], for example, might calculate λ from historical loss data and then use the Poisson formula to determine the probability of experiencing zero, one, two, or more claims per exposure unit. This output feeds directly into [[Definition:Premium | premium]] calculations, [[Definition:Reserving | reserve setting]], and [[Definition:Reinsurance | reinsurance]] structuring — particularly for [[Definition:Excess of loss reinsurance | excess-of-loss layers]] where the likelihood of multiple large losses hitting a single treaty period matters greatly. When [[Definition:Claims severity | severity]] must also be modeled, the Poisson distribution is often paired with a separate severity distribution in what actuaries call a [[Definition:Compound distribution | compound distribution]] framework.

💡 Accurate frequency modeling underpins virtually every pricing and capital decision an insurer makes, and the Poisson distribution's simplicity gives it enduring practical value. It allows [[Definition:Underwriting | underwriters]] and actuaries to quickly stress-test scenarios — such as what happens if frequency increases by 10 percent — and to communicate risk in terms stakeholders readily understand. While more complex models like the negative binomial distribution may be used when claims data shows greater variance than the Poisson assumes (a phenomenon known as [[Definition:Overdispersion | overdispersion]]), the Poisson remains the starting point for most frequency analyses and continues to be a core tool in [[Definition:Insurtech | insurtech]] platforms that automate [[Definition:Predictive analytics | predictive analytics]] for portfolio management.

'''Related concepts:'''
{{Div col|colwidth=20em}}
* [[Definition:Actuarial science]]
* [[Definition:Claims frequency]]
* [[Definition:Risk modeling]]
* [[Definition:Predictive analytics]]
* [[Definition:Loss distribution]]
* [[Definition:Compound distribution]]
{{Div col end}}

Definition:Poisson distribution - Revision history

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