Use our Pareto distribution calculator—a dedicated tool that computes both the probability density function (PDF) and the cumulative distribution function (CDF) of the Pareto distribution. By entering the shape parameter and scale parameter, the calculator provides key statistical measures, including the mean, variance, and skewness. It is an invaluable resource for researchers and practitioners who need to analyze data characterized by a Pareto distribution.

Pareto Distribution Calculator

Pareto Distribution Calculator

For parameters \( x_m \) (scale, minimum) and \( \alpha \) (shape), the PDF is:

$$ f(x) = \frac{\alpha \, x_m^\alpha}{x^{\alpha+1}}, \quad x \geq x_m. $$

and the CDF is:

$$ F(x) = 1 – \left(\frac{x_m}{x}\right)^\alpha, \quad x \geq x_m. $$

Step 1: Enter Parameters

Enter a positive number (e.g., 1)

Enter a positive number (e.g., 2)

Enter a number ≥ \( x_m \) (e.g., 2)

Pareto Distribution: $$ f(x) = \frac{\alpha \, x_m^\alpha}{x^{\alpha+1}} \quad \text{for } x \ge x_m. $$

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Understanding Pareto Distribution

The Pareto distribution function is a continuous probability distribution that models phenomena that become less common at larger scales. It's also known as the power law distribution. 

Characteristics 

  • The Pareto distribution is skewed and heavy-tailed.
  • It's often used to model income distribution and other financial variables.
  • It's named after economist Vilfredo Pareto.
  • When plotted on linear axes, it has a J-shaped curve.
  • When plotted in a log–log plot, it's represented by a straight line.

Formula

The formula for the Pareto distribution is f(x) = αx^(α + 1) for the probability density function (pdf) and F(x) = 1 - 1/x^α for the cumulative distribution function (cdf). 

Explanation

  • The Pareto distribution is a power law distribution that's often used to model income and other financial variables. 
  • The distribution is named after economist Vilfredo Pareto. 
  • The Pareto distribution is represented by a curved "long tail" on a linear scale. 
  • When plotted on a log-log graph, the Pareto distribution appears as a straight line with a negative gradient. 
  • The Pareto distribution is scale-invariant. This means that the ratio of values above a certain threshold is independent of the threshold. 

Parameters

  • The Pareto distribution has two parameters: α (“alpha”) and Xm. 
  • α is the shape parameter, which determines how the distribution is sloped. 
  • Xm is the minimum possible value of X. 

Real-world applications

The Pareto distribution is used in economics, other social sciences, and physical sciences. It's also known as the 80-20 rule, which states that 80% of outcomes come from 20% of causes. 

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