Insurance exposure ($K_j\mid \Theta \sim \mathrm{Pois}(w_j\Theta )$)

Insurance exposure measures the amount of opportunity for claims, such as the number of policy-years observed. With known positive insurance exposures $w_j$ and a common annual claim frequency $\Theta$, a simple model iswhere the claim counts $K_j$ are conditionally independent. The insurance exposures are fixed inputs here; the conditional independence assumption concerns the claim counts.

Put $W={\sum }_j{w}_j$ and $K={\sum }_j{K}_j$. Multiplying the conditional probability generating functions gives $K\mid \Theta \sim \mathrm{Poisson}(W\Theta )$. The conditional likelihood has the formAll dependence on $\theta$ is through the total claim count $K$ and total insurance exposure $W$; with the insurance exposures fixed, $K$ is a sufficient statistic. This is the exposure aggregation used in Poisson-gamma conjugacy with unequal exposures.

The normalized claim count $K_j/w_j$ has conditional expectation $\Theta$ and conditional variance $\Theta /w_j$. Thus its conditional expectation does not change with insurance exposure, while its conditional variance decreases. For example, two groups with $2$ and $3$ policy-years at annual frequency $0.4$ have expected claim counts $0.8$ and $1.2$; together they have a Poisson distribution claim count of mean $2$. The count model and aggregation follow from the Poisson distribution formula; the likelihood and example are derived here.^[ref]