# AP Statistics Curriculum 2007 Pareto

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+ | ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Pareto Distribution== | ||

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===Pareto Distribution=== | ===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. | '''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> | ||

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<br />'''Expectation''': The expected value of Pareto distributed random variable x is | <br />'''Expectation''': The expected value of Pareto distributed random variable x is | ||

- | :<math>E(X)=\frac{\alpha x_m}{\alpha-1} | + | :<math>E(X)=\frac{\alpha x_m}{\alpha-1}\mbox{ for }\alpha>1\!</math> |

<br />'''Variance''': The Pareto variance is | <br />'''Variance''': The Pareto variance is | ||

- | :<math>Var(X)=\frac{x_m^2 \alpha}{(\alpha-1)^2(\alpha-2)} | + | :<math>Var(X)=\frac{x_m^2 \alpha}{(\alpha-1)^2(\alpha-2)}\mbox{ for }\alpha>2\!</math> |

===Applications=== | ===Applications=== | ||

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*The areas burned in forest fires | *The areas burned in forest fires | ||

*The severity of large casualty losses for certain businesses, such as general liability, commercial auto, and workers compensation | *The severity of large casualty losses for certain businesses, such as general liability, commercial auto, and workers compensation | ||

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===Example=== | ===Example=== | ||

- | Suppose that the income of a certain population has a Pareto distribution with <math>\alpha=3</math> and <math>x_m=1000</math>. Compute the proportion of the population with incomes between 2000 and 4000. | + | Suppose that the income of a certain population has a Pareto distribution with <font size="3"><math>\alpha=3</math></font> and <font size="3"><math>x_m=1000</math></font>. Compute the proportion of the population with incomes between 2000 and 4000. |

We can compute this as follows: | We can compute this as follows: | ||

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The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html SOCR distributions] | The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html SOCR distributions] | ||

<center>[[Image:Pareto.jpg|600px]]</center> | <center>[[Image:Pareto.jpg|600px]]</center> | ||

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+ | <hr> | ||

+ | * SOCR Home page: http://www.socr.ucla.edu | ||

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+ | {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Pareto}} |

## Current revision as of 22:35, 18 July 2011

## Contents |

## General Advance-Placement (AP) Statistics Curriculum - Pareto Distribution

### 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

- SOCR Home page: http://www.socr.ucla.edu

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