Loss Random Variable

Study guide: Exam P-1

The loss random variable X represents the ground-up loss amount before any policy modifications such as Deductibles, Coinsurance Percentages, or Benefit Limits are applied.

• The Expected Value and Variance of the corresponding Payment Random Variable Y differ from those of X because of the truncation and censoring imposed by policy terms

Y = \begin{cases} 0 & X \leq d \\ X - d & X > d \end{cases}

\text{where } d = \text{ordinary deductible}

Expected Payment with Deductible Ground-up losses X \sim \text{Exp}(\theta = 1000). A policy has an ordinary deductible d = 500.

Answer The expected payment per loss is: E[Y] = E[\max(X - 500,\, 0)] = \int_{500}^{\infty}(x-500)\cdot\frac{1}{1000}e^{-x/1000}\,dx = 1000\,e^{-0.5} \approx 606.5