I'm trying to solve the following problem:
An insurance policy is written to cover a loss, $X$, where $X$ has a uniform distribution on $(0, 1000)$. At what level must a deductible be set in order for the expected payment to be $40\%$ of what it would be with no deductible?
$X$ = total loss
$D$ = deductible
$Y$ = insurance payment
I know that the insurance payment should be:
$$Y = \begin{cases}
0, & \text{for } X \leq D , \\
X – D, &\text{for } X > D.
\end{cases}
$$
So then the expected insurance payment is:
$$E[Y] = E[X – D|X >D] \cdot P[X>D]$ $
So that gives us:
$$E[Y] = \frac{1000+D}{2}\cdot\frac{1000-D}{1000}$$
After this I'm stuck.
Best Answer
With no deductible, the expected payment is $500$. $\frac{40}{100} \times 500 =200$. Let $d$ be the deductible. $\int\limits_{d}^{1000}\frac{(x-d)}{1000}dx=200$ so $\frac{(1000-D)^{2}}{2000}=200$, I hope I am not mistaken!