e. Compute the Gini index for the cases $$ \mathrm{L}(\mathrm{x})=x^{p} $$ and $$ p=1.1,1.3,2,3,4 $$

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To compute the **Gini index** for the cases where the Lorenz curve is defined by $$ \mathrm{L}(x) = x^{p} $$ with different values of $$ p $$, we can follow these steps:

### **Understanding the Lorenz Curve and Gini Index**

- **Lorenz Curve ($$ \mathrm{L}(x) $$)**: Represents the distribution of income or wealth within a population. Here, $$ \mathrm{L}(x) = x^{p} $$ where $$ x $$ ranges from 0 to 1.
  
- **Gini Index (G)**: Measures inequality within a distribution. It ranges from 0 (perfect equality) to 1 (maximal inequality).

### **Formula for Gini Index**

The Gini index can be calculated from the Lorenz curve using the following formula:

$$
G = 1 - 2 \int_{0}^{1} \mathrm{L}(x) \, dx
$$

For $$ \mathrm{L}(x) = x^{p} $$, the integral becomes:

$$
\int_{0}^{1} x^{p} \, dx = \frac{1}{p + 1}
$$

Substituting back into the formula for $$ G $$:

$$
G = 1 - 2 \left( \frac{1}{p + 1} \right) = \frac{p - 1}{p + 1}
$$

### **Calculations for Different Values of $$ p $$**

Let's compute the Gini index for each specified value of $$ p $$:

1. **For $$ p = 1.1 $$:**
   $$
   G = \frac{1.1 - 1}{1.1 + 1} = \frac{0.1}{2.1} \approx 0.0476 \quad \text{or} \quad 4.76\%
   $$

2. **For $$ p = 1.3 $$:**
   $$
   G = \frac{1.3 - 1}{1.3 + 1} = \frac{0.3}{2.3} \approx 0.1304 \quad \text{or} \quad 13.04\%
   $$

3. **For $$ p = 2 $$:**
   $$
   G = \frac{2 - 1}{2 + 1} = \frac{1}{3} \approx 0.3333 \quad \text{or} \quad 33.33\%
   $$

4. **For $$ p = 3 $$:**
   $$
   G = \frac{3 - 1}{3 + 1} = \frac{2}{4} = 0.5 \quad \text{or} \quad 50\%
   $$

5. **For $$ p = 4 $$:**
   $$
   G = \frac{4 - 1}{4 + 1} = \frac{3}{5} = 0.6 \quad \text{or} \quad 60\%
   $$

### **Summary of Gini Index Values**

| $$ p $$ | Gini Index ($$ G $$) | Percentage |
|---------|----------------------|------------|
| 1.1     | $$ \approx 0.0476 $$ | 4.76%      |
| 1.3     | $$ \approx 0.1304 $$ | 13.04%     |
| 2       | $$ \approx 0.3333 $$ | 33.33%     |
| 3       | $$ 0.5 $$             | 50%        |
| 4       | $$ 0.6 $$             | 60%        |

### **Interpretation**

- **Lower $$ p $$** values ($$ p = 1.1, 1.3 $$) indicate **less inequality** in the distribution.
- **Higher $$ p $$** values ($$ p = 2, 3, 4 $$) signify **greater inequality**.

This relationship shows that as $$ p $$ increases, the distribution becomes more unequal, which is reflected by the increasing Gini index.
The Gini index for $$ \mathrm{L}(x) = x^{p} $$ with $$ p = 1.1, 1.3, 2, 3, 4 $$ is calculated as follows: - $$ p = 1.1 $$: Gini index ≈ 4.76% - $$ p = 1.3 $$: Gini index ≈ 13.04% - $$ p = 2 $$: Gini index ≈ 33.33% - $$ p = 3 $$: Gini index = 50% - $$ p = 4 $$: Gini index = 60% As $$ p $$ increases, the Gini index increases, indicating greater inequality in the distribution.

e. Compute the Gini index for the cases \( \mathrm{L}(\mathrm{x})=x^{p} \) and \( p=1.1,1.3,2,3,4 \)

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