## Definition The ==Payment Random Variable== (denoted $Y$) represents the amount the insurer actually pays after applying all policy adjustments (deductible, coinsurance, benefit limit) to the loss random variable $X$. Analyzing $Y$ involves deriving its distribution, expected value, and variance from the distribution of $X$ and the policy terms. > [!example]- Losses $X$ are exponential with mean 1000. A policy has a deductible of 500 and no other adjustments. What is $E[Y]$? > $Y = (X - 500)^+$. By the memoryless property of the exponential: > $ E[Y] = E[X - 500 \mid X > 500] \cdot P(X > 500) = 1000 \cdot e^{-0.5} \approx 1000 \times 0.6065 = 606.53 $