The claim amount for medical expenses insurance is modeled with an exponential random variable with mean 20. The insurance company applies a deductible of 5. The payment function of the company is the amount of the claim. minus the deductible, when this amount is positive and zero otherwise. What is the expected payment that you will have to make the company?
Let $R$ be the random variable that represents the claim amount then $$f_R(r)=\frac{1}{20}e^{-\frac{1}{20}r}\mathbb{I}_{[0,\infty)}(r)$$ Then let $P$ be the payment function of the firm then $$P(r)=\left\{ \begin{array}{lcc} \frac{1}{20}e^{-\frac{1}{20}r}-5 \hspace{1cm} r<-40ln(10)\\0 \hspace{1cm} In\hspace{.2cm} another\hspace{.2cm} case \end{array}\right.$$
It is what I have tried but I suspect that it is wrong, can someone help me?
Here $P = (R-5)\cdot\mathsf 1_{(5,\infty)}(R)$. It follows that $$ \mathbb P(P=0) = \mathbb P(R<5) = \int_0^5 \frac1{20} e^{-\frac1{20} r}\ \mathsf dr = 1-e^{-\frac14}, $$ and for $r>0$ $$ \mathbb P(P>r) = \mathbb P(R>r+5) = \int_{r+5}^\infty\frac1{20} e^{-\frac1{20} r}\ \mathsf dr = e^{-\frac{5+r}{20}}. $$ From this we may compute $$ \mathbb E[P] = \int_0^\infty \mathbb P(P>r)\ \mathsf dr = \int_0^\infty e^{-\frac{5+r}{20}}\ \mathsf dr = 20 e^{\frac14}. $$