AP Statistics Curriculum 2007 Pareto - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T17:50:35Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

JayZzz at 22:41, 18 July 2011

2011-07-18T22:41:03Z

JayZzz at 22:40, 18 July 2011

2011-07-18T22:40:49Z

TracyTam: /* Pareto Distribution */

2011-07-11T21:24:28Z

‎Pareto Distribution

TracyTam: /* Applications */

2011-07-11T21:10:12Z

‎Applications

TracyTam: /* Pareto Distribution */

2011-07-11T21:04:27Z

‎Pareto Distribution

TracyTam: /* Example */

2011-07-11T20:54:47Z

‎Example

TracyTam: /* Pareto Distribution */

2011-07-11T20:54:17Z

‎Pareto Distribution

TracyTam: 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 …'

2011-07-11T19:39:12Z

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 …'

New page

===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~Pareto(<math>x_m,\alpha</math>), the Pareto probability density function is given by

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

where
*<math>x_m</math> is the minimum possible value of X
*<math>\alpha</math> is a positive parameter which determines the concentration of data towards the mode
*x is a random variable (<math>x>x_m</math>)

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

:<math>1-(\frac{x_m}{x})^\alpha</math>

where
*<math>x_m</math> is the minimum possible value of X
*<math>\alpha</math> is a positive parameter which determines the concentration of data towards the mode
*x is a random variable (<math>x>x_m</math>)

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

:<math>M(t)=\alpha(-x_m t)^\alpha\Gamma(-\alpha,-x_m t)\!</math>

where
*<math>\Gamma(-\alpha,-x_m t)=\int_{-x_m t}^\infty t^{-\alpha-1}e^{-t}dt</math>

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

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

 '''Variance''': The Pareto variance is

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

===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 <math>\alpha=3</math> and <math>x_m=1000</math>. Compute the proportion of the population with incomes between 2000 and 4000.

We can compute this as follows:

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

The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html SOCR distributions]
<center>[[Image:Pareto.jpg|600px]]</center>

← Older revision		Revision as of 17:50, 3 March 2020
Line 64:		Line 64:
	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php/AP_Statistics_Curriculum_2007_Pareto}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php/AP_Statistics_Curriculum_2007_Pareto}}

← Older revision		Revision as of 22:41, 18 July 2011
Line 64:		Line 64:
	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.stat.ucla.edu/socr/index.php/~~AP_Statistics_Curriculum_2007_Chi-Square~~}}	+	{{translate\|pageName=http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Pareto}}

@@ Line 2: / Line 2: @@
 '''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>

@@ Line 29: / Line 29: @@
 <br />'''Expectation''': The expected value of Pareto distributed random variable x is
-:<math>E(X)=\frac{\alpha x_m}{\alpha-1}</math> for <math>\alpha>1\!</math>
+:<math>E(X)=\frac{\alpha x_m}{\alpha-1}\mbox{ for }\alpha>1\!</math>
 <br />'''Variance''': The Pareto variance is
-:<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)}\mbox{ for }\alpha>2\!</math>
 ===Applications===

@@ Line 2: / Line 2: @@
 '''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 X~Pareto(<math>x_m,\alpha</math>), the Pareto probability density function is given by
+<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>
 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 />'''Cumulative density function''': The Pareto cumulative distribution function is given by
@@ Line 16: / Line 16: @@
 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
@@ Line 25: / Line 25: @@
 where
-*<math>\Gamma(-\alpha,-x_m t)=\int_{-x_m t}^\infty t^{-\alpha-1}e^{-t}dt</math>
+*<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
@@ Line 34: / Line 34: @@
 :<math>Var(X)=\frac{x_m^2 \alpha}{(\alpha-1)^2(\alpha-2)}</math> for <math>\alpha>2\!</math>
 ===Applications===

@@ Line 1: / Line 1: @@
 ===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.
@@ Line 58: / Line 60: @@
 The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Pareto_Distribution.html SOCR distributions]
 <center>[[Image:Pareto.jpg|600px]]</center>

@@ Line 48: / Line 48: @@
 *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===

@@ Line 51: / Line 51: @@
 ===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: