The question is:
The cumulative distribution function for health care costs experienced by a policyholder is modeled by the function
\begin{align*}
F(x)=\left\{\begin{matrix}
1-\mathit{e}^{-\frac{x}{100}} & &x>0 \\
0& &\text{otherwise}
\end{matrix}\right..
\end{align*}
The policy has a deductible of 20. An insurer reimburses the policyholder for 100$\%$ of health are costs between 20 and 120 less the deductible. Health care costs above 120 are reimbursed at 50$\%$.Let $G$ be the cumulative distribution function of reimbursements given that the reimbursement is positive. Calculate $G(115)$.
Hello, I am trying to understand a solution I found online that gave
$G(y)=\left\{\begin{matrix}
0 &x<20 \\
x-20 &20\leq x\leq 120 \\
0.5(x-120) & x>120
\end{matrix}\right.$
where Y is the reimbursement and X is the cost.
I understand everything up to the third line of G(y). Why is it subtracting 120 from x?
Best Answer
To solve the problem (I guess it's an SOA (P) question) efficiently, you don't need to spell out $G(y)$ for all $y > 0$. Instead, by law of total probability, you can solve $G(115)$ directly as follows:
\begin{align} & G(115) = P(Y \leq 115) \\ =& P(Y \leq 115 | X \leq 20)P(X \leq 20) + \\ & P(Y \leq 115 | 20 < X \leq 120)P(20 < X \leq 120) + \\ & P(Y \leq 115 | X > 120) P(X > 120) \\ =& P(X \leq 20) + \\ & P(X - 20 \leq 115|20 < X \leq 120)P(20 < X \leq 120) + \\ & P(100 + 0.5(X - 120) \leq 115 | X > 120)P(X > 120) \\ =& P(X \leq 20) + P(X \leq 135 | 20 < X \leq 120)P(20 < X \leq 120) + P(X \leq 150 | X > 120)P(X > 120) \\ =& P(X \leq 20) + P(20 < X \leq 120) + P(120 < X \leq 150) \\ =& P(X \leq 150) = 1 -e^{-150/100} = 1 - e^{-1.5} = 0.7769. \end{align}