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

Insurance exposure scales the opportunity for claims: $w_j$ policy-years produce expected count $w_j\Theta$ at common annual frequency $\Theta$. Conditionally independent insurance exposures add, so the total count and total insurance exposure are sufficient statistics in a Poisson distribution frequency model. A yearly claim average has conditional expectation independent of the number of policies, but its conditional variance falls with that number.