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Pareto Distribution


The Pareto Distribution is a continuous probability distribution used to describe many empirical observations. It was originally used by the Italian economist Vilfredo Pareto who used it to model the distribution of wealth in cities.

Density Curve of the Pareto Distribution:

The probability distribution function is given as:

PDF of pareto distribution

where xm is the minimum possible value of x and α is the shape parameter.

The probability density curve when plotted looks like a rectangular hyperbola with the graph becoming asymptotic to the x-axis as x tends to infinity. The graph is “long-tailed”.

Graph of PDF of pareto distribution

Mean and Variance of the Pareto Distribution:

The mean and variance of the Pareto distribution can be calculated as follows,

Formula for mean and variance of the Pareto distribution

Pareto Principle:

The Pareto principle or the 80-20 rules states that wealth is distributed in such a way that 20% of the population has 80% of the riches. It has been observed empirically that richer people tend to own a greater amount of wealth in comparison to the rest of the population.

So a substantial portion of the total wealth(80% of the total wealth) lies in the hands of the extremely rich(the top 20%). The bottom 80% of the people own only 20% of the total wealth.

Applications of the Pareto Distribution:

The Pareto distribution is widely used in:

  1. The modelling of wealth distribution in countries.
  2. Modelling the spread of population.
  3. Modelling the gains made in the stock market.

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