= Insurance exposure
{title2=$K_j\mid\Theta\sim\operatorname{Pois}(w_j\Theta)$}
<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 is
$$
K_j\mid\Theta\sim\operatorname{Poisson}(w_j\Theta),
$$
where the <claim counts> $K_j$ are <conditionally independent>. The <insurance exposures> are fixed inputs here; the <conditional independence> assumption concerns the <claim counts>.
Put $W=\sum_jw_j$ and $K=\sum_jK_j$. Multiplying the conditional <probability generating functions> gives $K\mid\Theta\sim\operatorname{Poisson}(W\Theta)$. The conditional <likelihood> has the form
$$
L(\theta)=\left(\prod_j\frac{w_j^{k_j}}{k_j!}\right)\theta^K e^{-W\theta}.
$$
All 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.<Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik>[Section 3.1.5, Poisson distribution; exposure likelihood and aggregation derived here]{ref}
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