For a certain health insurance policy, losses are uniformly distributed on the interval [0, [math]b[/math]]. The policy has a deductible of 180 and the expected value of the unreimbursed portion of a loss is 144.

Calculate [math]b[/math].

• 236
• 288
• 388
• 450
• 468

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: D

If [math]L[/math] is the loss, the unreimbursed loss, [math]X[/math] is

[[math]] X = \begin{cases} L, \quad L \leq 180 \\ 180, L \gt 180 \end{cases} [[/math]]

The expected unreimbursed loss is

[[math]] \begin{align*} 144 &= \operatorname{E}(X) = \int_0^{100} l[f(l)] dl + 180 P(L \gt 180) = \int_0^{180} l \frac{1}{b} dl + 180 \frac{b-180}{b} \\ &= \frac{l^2}{2b} |_0^{180} + 180 - \frac{180^2}{b} = \frac{180^2}{2b} + 180 - \frac{180^2}{b} \\ 144b &= 180^2 /2 + 180b - 180^2 \\ 16200 &= 36b \\ b &= 450. \end{align*} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.