# AP Statistics Curriculum 2007 Pareto

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'''Definition''': Pareto distribution is a skewed, heavy-tailed distribution that is sometimes used to model that distribution of incomes. The basis of the distribution is that a high proportion of a population has low income while only a few people have very high incomes. | '''Definition''': Pareto distribution is a skewed, heavy-tailed distribution that is sometimes used to model that distribution of incomes. The basis of the distribution is that a high proportion of a population has low income while only a few people have very high incomes. | ||

- | <br />'''Probability density function''': For <math>X\sim Pareto(x_m,\alpha)\!</math>, the Pareto probability density function is given by | + | <br />'''Probability density function''': For <math>X\sim \operatorname{Pareto}(x_m,\alpha)\!</math>, the Pareto probability density function is given by |

:<math>\frac{\alpha x_m^\alpha}{x^{\alpha+1}}</math> | :<math>\frac{\alpha x_m^\alpha}{x^{\alpha+1}}</math> |

## Revision as of 21:18, 11 July 2011

### Pareto Distribution

**Definition**: Pareto distribution is a skewed, heavy-tailed distribution that is sometimes used to model that distribution of incomes. The basis of the distribution is that a high proportion of a population has low income while only a few people have very high incomes.

**Probability density function**: For , the Pareto probability density function is given by

where

*x*_{m}is the minimum possible value of X- α is a positive parameter which determines the concentration of data towards the mode
- x is a random variable (
*x*>*x*_{m})

**Cumulative density function**: The Pareto cumulative distribution function is given by

where

*x*_{m}is the minimum possible value of X- α is a positive parameter which determines the concentration of data towards the mode
- x is a random variable (
*x*>*x*_{m})

**Moment generating function**: The Pareto moment-generating function is

where

**Expectation**: The expected value of Pareto distributed random variable x is

**Variance**: The Pareto variance is

### Applications

The Pareto distribution is sometimes expressed more simply as the “80-20 rule”, which describes a range of situations. In customer support, it means that 80% of problems come from 20% of customers. In economics, it means 80% of the wealth is controlled by 20% of the population. Examples of events that may be modeled by Pareto distribution include:

- The sizes of human settlements (few cities, many villages)
- The file size distribution of Internet traffic which uses the TCP protocol (few larger files, many smaller files)
- Hard disk drive error rates
- The values of oil reserves in oil fields (few large fields, many small fields)
- The length distribution in jobs assigned supercomputers (few large ones, many small ones)
- The standardized price returns on individual stocks
- The sizes of sand particles
- The sizes of meteorites
- The number of species per genus
- The areas burned in forest fires
- The severity of large casualty losses for certain businesses, such as general liability, commercial auto, and workers compensation

### Example

Suppose that the income of a certain population has a Pareto distribution with α = 3 and *x*_{m} = 1000. Compute the proportion of the population with incomes between 2000 and 4000.

We can compute this as follows:

The figure below shows this result using SOCR distributions