Find the Gini index for this function. Recall that \( \mathrm{L}(\mathrm{x})=\frac{5 \mathrm{x}^{2}}{6}+\frac{\mathrm{x}}{6} \). \( \mathrm{G}=\square \) (Type an integer or a fraction.)

To find the Gini index for the given function \( \mathrm{L}(\mathrm{x}) = \frac{5\mathrm{x}^2}{6} + \frac{\mathrm{x}}{6} \), we'll follow these steps: 1. **Understand the Lorenz Curve**: - The function \( \mathrm{L}(\mathrm{x}) \) represents the Lorenz curve, which is used to calculate the Gini index. 2. **Gini Index Formula**: - The Gini index (\( \mathrm{G} \)) is calculated as: \[ \mathrm{G} = 1 - 2 \times \text{Area under the Lorenz curve} \] 3. **Calculate the Area Under the Lorenz Curve**: - Integrate \( \mathrm{L}(\mathrm{x}) \) from 0 to 1: \[ \int_0^1 \left( \frac{5\mathrm{x}^2}{6} + \frac{\mathrm{x}}{6} \right) d\mathrm{x} \] - Compute the integral: \[ \int_0^1 \frac{5\mathrm{x}^2}{6} d\mathrm{x} = \frac{5}{6} \times \frac{1}{3} = \frac{5}{18} \] \[ \int_0^1 \frac{\mathrm{x}}{6} d\mathrm{x} = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \] - Add the two results: \[ \frac{5}{18} + \frac{1}{12} = \frac{10}{36} + \frac{3}{36} = \frac{13}{36} \] 4. **Compute the Gini Index**: \[ \mathrm{G} = 1 - 2 \times \frac{13}{36} = 1 - \frac{26}{36} = \frac{10}{36} = \frac{5}{18} \] **Answer:** \(\boxed{\dfrac{5}{18}}\)