Pareto Distribution

📈 Pareto Distribution in Python

The Pareto Distribution is a heavy-tailed continuous probability distribution often used in economics, finance, and social sciences.
It is famous for modeling wealth distribution, income, or city sizes, where a small number of occurrences account for the majority of the effect (the 80/20 rule).

1. Characteristics of Pareto Distribution

Continuous distribution
Parameters:
- a → shape parameter (α > 0)
- size → number of samples
Probability Density Function (PDF):

$x_m^a / x^{a+1}, \quad x \ge x_m$

x_m → minimum possible value (default in NumPy = 1)
Heavy-tailed (large values are possible)

Applications:

Wealth or income distribution
File sizes in the internet
Population sizes of cities

2. Generate Pareto Data Using NumPy

Output (example):

Values ≥ 0
Can shift using x_m + data if minimum value other than 1 is required

3. Visualize Pareto Distribution

Histogram is right-skewed
Few very large values → heavy tail

4. Shifted Pareto Distribution

Shift ensures minimum value ≥ xm

5. Compare Different Shape Parameters

data1 = np.random.pareto(a=2, size=1000)
data2 = np.random.pareto(a=5, size=1000)

sns.histplot(data1, bins=50, color='red', label='a=2', kde=True, alpha=0.5)
sns.histplot(data2, bins=50, color='blue', label='a=5', kde=True, alpha=0.5)
plt.title("Pareto Distribution Comparison")
plt.legend()
plt.show()

data1 = np.random.pareto(a=2, size=1000)

data2 = np.random.pareto(a=5, size=1000)

sns.histplot(data1, bins=50, color='red', label='a=2', kde=True, alpha=0.5)

sns.histplot(data2, bins=50, color='blue', label='a=5', kde=True, alpha=0.5)

plt.title("Pareto Distribution Comparison")

plt.legend()

plt.show()

Smaller a → heavier tail (more extreme values)
Larger a → distribution more concentrated near 0

🧠 Summary Table

Function	Parameters	Description
`np.random.pareto()`	a, size	Generates Pareto random numbers
`a`	Shape parameter	Controls tail heaviness
`size`	Number of samples	Output array size
`xm`	Minimum value	Can shift data by adding xm

🎯 Practice Exercises

Generate 1000 Pareto random numbers with a=2 and plot histogram with KDE.
Compare Pareto distributions for a=2 vs a=5.
Shift Pareto data to have minimum value 10 and visualize.

Pareto Distribution