Mean Excess Loss 329Fe1

1. **State the problem:** We have a piecewise density function $$f(x) = 0.05$$ for $$0 < x \leq 5$$ and $$f(x) = 0.15$$ for $$5 < x \leq 10$$. We want to calculate the mean excess loss for deductibles $$d=10$$ and $$d=2.6$$. 2. **Recall the mean excess loss formula:** The mean excess loss at deductible $$d$$ is given by: $$e(d) = E[(X - d)^+] = \frac{\int_d^{\infty} (x - d) f(x) dx}{P(X > d)}$$ where $$P(X > d) = 1 - F(d)$$ and $$F(d)$$ is the cumulative distribution function. 3. **Calculate the cumulative distribution function (CDF) $$F(x)$$:** - For $$0 < x \leq 5$$: $$F(x) = \int_0^x 0.05 \, dt = 0.05x$$ - For $$5 < x \leq 10$$: $$F(x) = F(5) + \int_5^x 0.15 \, dt = 0.05 \times 5 + 0.15(x - 5) = 0.25 + 0.15(x - 5)$$ 4. **Calculate mean excess loss for deductible $$d=10$$:** - Since $$d=10$$ is the upper limit of the support, $$P(X > 10) = 0$$. - Therefore, $$e(10) = 0$$ because no loss exceeds 10. 5. **Calculate mean excess loss for deductible $$d=2.6$$:** - Calculate $$P(X > 2.6) = 1 - F(2.6)$$. Since $$2.6 \leq 5$$, $$F(2.6) = 0.05 \times 2.6 = 0.13$$ So, $$P(X > 2.6) = 1 - 0.13 = 0.87$$ - Calculate numerator $$\int_{2.6}^{10} (x - 2.6) f(x) dx$$ split into two intervals: For $$2.6 < x \leq 5$$: $$\int_{2.6}^5 (x - 2.6) 0.05 dx$$ For $$5 < x \leq 10$$: $$\int_5^{10} (x - 2.6) 0.15 dx$$ 6. **Evaluate the first integral:** $$\int_{2.6}^5 (x - 2.6) 0.05 dx = 0.05 \int_{2.6}^5 (x - 2.6) dx = 0.05 \left[ \frac{(x - 2.6)^2}{2} \right]_{2.6}^5$$ Calculate: $$5 - 2.6 = 2.4$$ So, $$= 0.05 \times \frac{(2.4)^2}{2} = 0.05 \times \frac{5.76}{2} = 0.05 \times 2.88 = 0.144$$ 7. **Evaluate the second integral:** $$\int_5^{10} (x - 2.6) 0.15 dx = 0.15 \int_5^{10} (x - 2.6) dx = 0.15 \left[ \frac{(x - 2.6)^2}{2} \right]_5^{10}$$ Calculate: $$10 - 2.6 = 7.4$$ $$5 - 2.6 = 2.4$$ So, $$= 0.15 \times \frac{7.4^2 - 2.4^2}{2} = 0.15 \times \frac{54.76 - 5.76}{2} = 0.15 \times \frac{49}{2} = 0.15 \times 24.5 = 3.675$$ 8. **Sum the integrals for numerator:** $$0.144 + 3.675 = 3.819$$ 9. **Calculate mean excess loss for $$d=2.6$$:** $$e(2.6) = \frac{3.819}{0.87} \approx 4.39$$ **Final answers:** - Mean excess loss for deductible $$10$$ is $$0$$. - Mean excess loss for deductible $$2.6$$ is approximately $$4.39$$.