Problem: Auto Claim amounts, in thousands, are modeled by a random variable with density function $f_x(x)=xe^{-x}$ for $x\ge 0$. The company expects to pay 100 claims if there is no deductible. How many claims does the company expect to pay if the company decides to introduce a deductible of 1000?
My Solution: $F_x(1)=\int_0^1 xe^{-x} dx = 1-e^{-1}$. Thus, $s_x(1)=1-F_x(1)=e^{-1}$. So the company expects to pay $e^{-1}*100\approx 37$ claims.
The solution given in my study manual for this year's P/1 Exam says that $s_x(1)=2e^{-1}$ and that the company expects to pay $2e^{-1}*100 \approx 74$ claims.
I understand their proof, but can't see where mine has failed. Can someone please point out to me where I went wrong?
Thanks!
Your calculation is incorrect. $$F_X(1) = \int_{x=0}^1 xe^{-x} \, dx = \left[-(1+x)e^{-x}\right]_{x=0}^1 = -2e^{-1} + 1.$$ Since you did not show how you arrived at $1 - e^{-1}$, I cannot furnish any more details regarding where you made your error.