Compound Poisson Distribution¶

Definition¶

A compound Poisson loss is $\displaystyle S=\sum_{i=1}^NX_i$ with $N\sim\operatorname{Pois}(\lambda)$$\mathrm{Pois}(\lambda )$, $\lambda\ge0$$0$, and i.i.d. nonnegative severities independent of $N$. The empty sum is zero.

Interpretation¶

Poisson frequency makes aggregate losses particularly tractable. Even with continuous positive severities, aggregate loss includes a probability of no claims at all.

The figure plots the Example's loss $S=2N$ with $\lambda=3$$3$, so $\mathbb{P}(S=2k)=\mathrm{e}^{-3}3^k/k!$${\mathrm{e}}^{-3}{3}^k/k!$. The orange bar at zero is the no-claim probability $\mathbb{P}(S=0)=\mathrm{e}^{-\lambda}=\mathrm{e}^{-3}\approx0.05$$0.05$, present even though every claim is positive. The mass sits only on even values, and the mean $6=\lambda\mathbb{E}[X]$ and variance $12=\lambda\mathbb{E}[X^2]$ follow from the Key Results; a Poisson variable with mean 6 would have variance 6.

This specializes the compound risk distribution by choosing the count from the Poisson distribution. Since its count mean and variance both equal $\lambda$, the general random-sum variance simplifies to $\lambda\mathbb{E}[X^2]$.

Key Results¶

• $\mathbb{E}[S]=\lambda\mathbb{E}[X]$ and $\operatorname{Var}(S)=\lambda\mathbb{E}[X^2]$ when finite.
• The moment generating function (MGF) satisfies $M_S(t)=\exp(\lambda(M_X(t)-1))$ where finite.
• With strictly positive severity, $\mathbb{P}(S=0)=\mathrm{e}^{-\lambda}$${\mathrm{e}}^{-\lambda }$.
• Claim counts for disjoint severity categories are independent Poisson variables with rates $\lambda p_j$ (with $p_j$ the probability that a claim falls in category $j$); this uses the Poisson splitting result.

Examples¶

If each claim is 2 and $\lambda=3$$3$, then $S=2N$ has mean 6 and variance 12. Its support is $0,2,4,\ldots$, so it is not an ordinary Poisson variable.

Prerequisites¶

• Compound Risk Distribution
• Poisson Distribution

Related Concepts¶

• Superposition of Compound Poisson Risks
• Compound Poisson Process
• Poisson Splitting
• Collective Risk Model
• Cumulant Generating Function
• Maximum Claim in a Portfolio
• Risk Orders Under Aggregation

Sources¶

• Risk Insurance · Lecture 4

Aliases: compound Poisson aggregate loss, compound Poisson, compound Poisson random variable Keywords: Poisson frequency model, M_S(t) = exp(λ(M_X(t) - 1)), Var(S) = λ E[X^2], no-claim probability e^(-λ), Poisson splitting by claim category Symbols: lambda, sum

Last updated: 17 October 2026 21:53