AP Statistics Curriculum 2007 Pareto

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(Created page with '===Pareto Distribution=== '''Definition''': Pareto distribution is a skewed, heavy-tailed distribution that is sometimes used to model that distribution of incomes. The basis of …')
(Pareto Distribution)
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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.
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<br />'''Probability density function''': For X~Pareto(<math>x_m,\alpha</math>), the Pareto probability density function is given by
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<br />'''Probability density function''': For <math>X\sim 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>
where
where
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*<math>x_m</math> is the minimum possible value of X
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*<font size="3"><math>x_m</math></font> is the minimum possible value of X
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*<math>\alpha</math> is a positive parameter which determines the concentration of data towards the mode
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*<font size="3"><math>\alpha</math></font> is a positive parameter which determines the concentration of data towards the mode
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*x is a random variable (<math>x>x_m</math>)
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*x is a random variable (<font size="3"><math>x>x_m</math></font>)
<br />'''Cumulative density function''': The Pareto cumulative distribution function is given by
<br />'''Cumulative density function''': The Pareto cumulative distribution function is given by
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where
where
-
*<math>x_m</math> is the minimum possible value of X
+
*<font size="3"><math>x_m</math></font> is the minimum possible value of X
-
*<math>\alpha</math> is a positive parameter which determines the concentration of data towards the mode
+
*<font size="3"><math>\alpha</math></font> is a positive parameter which determines the concentration of data towards the mode
-
*x is a random variable (<math>x>x_m</math>)
+
*x is a random variable (<font size="3"><math>x>x_m</math></font>)
<br />'''Moment generating function''': The Pareto moment-generating function is  
<br />'''Moment generating function''': The Pareto moment-generating function is  
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where
where
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*<math>\Gamma(-\alpha,-x_m t)=\int_{-x_m t}^\infty t^{-\alpha-1}e^{-t}dt</math>
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*<math>\textstyle\Gamma(-\alpha,-x_m t)=\int_{-x_m t}^\infty t^{-\alpha-1}e^{-t}dt</math>
<br />'''Expectation''': The expected value of Pareto distributed random variable x is  
<br />'''Expectation''': The expected value of Pareto distributed random variable x is  
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:<math>Var(X)=\frac{x_m^2 \alpha}{(\alpha-1)^2(\alpha-2)}</math> for <math>\alpha>2\!</math>
:<math>Var(X)=\frac{x_m^2 \alpha}{(\alpha-1)^2(\alpha-2)}</math> for <math>\alpha>2\!</math>
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===Applications===
===Applications===

Revision as of 20:46, 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 X\sim Pareto(x_m,\alpha)\!, the Pareto probability density function is given by

\frac{\alpha x_m^\alpha}{x^{\alpha+1}}

where

  • xm 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 > xm)


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

1-(\frac{x_m}{x})^\alpha

where

  • xm 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 > xm)


Moment generating function: The Pareto moment-generating function is

M(t)=\alpha(-x_m t)^\alpha\Gamma(-\alpha,-x_m t)\!

where

  • \textstyle\Gamma(-\alpha,-x_m t)=\int_{-x_m t}^\infty t^{-\alpha-1}e^{-t}dt


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

E(X)=\frac{\alpha x_m}{\alpha-1} for \alpha>1\!


Variance: The Pareto variance is

Var(X)=\frac{x_m^2 \alpha}{(\alpha-1)^2(\alpha-2)} for \alpha>2\!

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 xm = 1000. Compute the proportion of the population with incomes between 2000 and 4000.

We can compute this as follows:

P(2000\le X\le 4000)=\sum_{x=2000}^{4000}\frac{3\times 1000^3}{x^{3+1}}=0.109375

The figure below shows this result using SOCR distributions

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