How to Solve Insurance Risk Variables — Insurance — coinsurance, inflation, and deductible Questions on Exam P

For past claim \(X \sim \operatorname{Exponential}(\lambda)\), survival function is \(S_X(x) = e^{-\lambda x}\). Given \(\Pr(X<1200) = 1 - S_X(1200) = 0.20\), survival at 1200 is \(S_X(1200) = e^{-1200\lambda} = 1 - 0.20 = 0.80\). For inflated claim \(Y = 1.5X\), survival at threshold \(y\) is \(\Pr(Y > y) = \Pr(1.5X > y) = \Pr(X > y/1.5) = S_X(y/1.5) = e^{-\lambda (y/1.5)} = (e^{-1200\lambda})^{\frac{y/1.5}{1200}} = (0.80)^{y/1800}\). For threshold \(y = 1200\), survival is \(\Pr(Y > 1200) = (0.80)^{1200/1800} = (0.80)^{2/3}\). Taking the complement yields \(\Pr(Y < 1200) = 1 - (0.80)^{2/3}\). Therefore, \(\Pr(Y<1200)\) is \(0.1382\). This is the correct option.