Pareto Distribution

The Pareto distribution, also known as the Bradford distribution, is used to model variables that follow Pareto's 80-20 principle: 80% of resources are owned by 20% of the population, etc.

The Pareto distribution has a location parameter which must be strictly greater than 0 and corresponds to the smallest possible value of the variable. It also has a scale parameter, also positive, that determines how fast the distribution drops off from the smallest value. The probability density function (PDF) is:

$$f(x;x_m,\alpha )=\frac{\alpha x_m^{\alpha }}{x^{\alpha +1}}\text{for}x\ge x_m,\alpha >0$$

The cumulative distribution function (CDF) is:

$$F(x;x_m,\alpha )=1-{\left(\frac{x_m}{x}\right)}^{\alpha }\text{for}x\ge x_m,\alpha >0$$

• Economics uses the Pareto distribution to model the distribution of wealth and income in societies.

• Internet traffic analysis employs it to model file sizes and transmission times.

• Reliability engineering uses it to model failure times of components.

• Social networks utilize it to analyze node degrees and connection patterns.

The Pareto distribution exhibits heavy tail behavior with infinite moments of order greater than or equal to $\alpha$.

Notable properties include:

• The distribution exhibits scale-free behavior.

• The hazard rate is monotone decreasing.

• All moments of order $k\ge \alpha$ are infinite.

• If X follows a Pareto distribution, then ln(X/x_m) follows an exponential distribution.

• The Pareto distribution is a special case of the generalized Pareto distribution.

• It is related to the Zipf distribution through a power law relationship.

The Pareto distribution is implemented by the ParetoDistribution class. It has one constructor that takes two arguments. The first argument is the location parameter, and corresponds to the mode of the probability density function. The second argument is the shape parameter.

The following constructs the Pareto distribution with location parameter 6.8 and scale parameter 1.8:

var pareto = new ParetoDistribution(6.8, 1.8);

Dim pareto = New ParetoDistribution(6.8, 1.8)

No code example is currently available or this language may not be supported.

let pareto = ParetoDistribution(6.8, 1.8)

The ParetoDistribution class has two specific properties, ShapeParameter and ScaleParameter, which return the shape parameter and the scale parameter of the distribution.

ParetoDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. The second and third parameters are the location and scale parameters of the distribution.

var random = new Pcg32();
double sample = ParetoDistribution.Sample(random, 6.8, 1.8);

Dim random = New Pcg32()
Dim sample = ParetoDistribution.Sample(random, 6.8, 1.8)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = ParetoDistribution.Sample(random, 6.8, 1.8)

The above example uses the Pcg32 class to generate uniform random numbers.

For more information on the Pareto distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 20. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 34. Wiley Series in Probability and Statistics.

• Arnold, B.C. (2015). Pareto Distributions. Chapman and Hall/CRC.

• Newman, M.E.J. (2005). Power laws, Pareto distributions and Zipf's law. Contemporary Physics, 46(5), 323-351.

• Reed, W.J. (2001). The Pareto, Zipf and other power laws. Economics Letters, 74(1), 15-19.

Other Resources