Pareto distribution

The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto,[1] (Italian: [paˈreːto] US: /pəˈrt/ pə-RAY-toh),[2] is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial, and many other types of observable phenomena. Originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population.[3][4] The Pareto principle or "80-20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value (α) of log45  1.16 precisely reflect it. Empirical observation has shown that this 80-20 distribution fits a wide range of cases, including natural phenomena[5] and human activities.[6]

Pareto Type I
Probability density function

Pareto Type I probability density functions for various with As the distribution approaches where is the Dirac delta function.
Cumulative distribution function

Pareto Type I cumulative distribution functions for various with
Parameters scale (real)
shape (real)
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF does not exist
CF
Fisher information

Right: