$Y = \alpha\,\min\!\bigl((X-d)_+,\; u\bigr)$ $\text{where } d = \text{deductible},\; u = \text{benefit limit},\; \alpha = \text{coinsurance}$ The Payment Random Variable ($Y$) is the amount the insurer actually pays on a claim, derived from the loss random variable $X$ after applying all policy provisions (deductible, coinsurance, and benefit limit). $Y$ is a transformed version of $X$ and has a mixed distribution: a probability mass at $Y = 0$ (when $X \leq d$) and a continuous or discrete component for positive payments. Its mean and variance differ from those of $X$ due to the truncation and censoring imposed by the policy. > [!example]- Full Payment Function with Three Provisions {💡 Example} > A policy has deductible $d = 200$, coinsurance $\alpha = 0.75$, and benefit limit $u = 900$. If $X = 1500$, find the payment $Y$. > > > [!answer]- Answer > > Step 1 — Apply deductible: $X - d = 1500 - 200 = 1300$. > > Step 2 — Apply benefit limit: $\min(1300, 900) = 900$. > > Step 3 — Apply coinsurance: > > $Y = 0.75 \times 900 = 675$ > > The insurer pays \$675. The insured absorbs \$200 (deductible) + \$400 (excess above limit) = \$600, and co-pays $0.25 \times 900 = \$225$, totaling \$825.