AP Statistics Curriculum 2007 Cauchy - Revision history

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JayZzz: /* Cauchy Distribution */

2011-07-18T22:44:57Z

‎Cauchy Distribution

JayZzz: /* Cauchy Distribution */

2011-07-07T04:35:27Z

‎Cauchy Distribution

JayZzz at 04:21, 7 July 2011

2011-07-07T04:21:25Z

JayZzz at 04:21, 7 July 2011

2011-07-07T04:21:08Z

JayZzz at 22:04, 30 June 2011

2011-06-30T22:04:33Z

JayZzz: Created page with '== General Advance-Placement (AP) Statistics Curriculum - Cauchy Distribution== === Cauchy Distribution === The standard Cauchy distribution i…'

2011-06-30T22:04:18Z

Created page with '== General Advance-Placement (AP) Statistics Curriculum - Cauchy Distribution== === Cauchy Distribution === The standard Cauchy distribution i…'

New page

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

=== Cauchy Distribution ===
The standard Cauchy distribution is derived from the ratio of two independent Normal distributions. If X ~ N(0,1), and Y ~ N(0,1), then <math>\tfrac{X}{Y} \sim Cauchy(0,1)</math>

The Cauchy distribution is important in physics (where it’s known as the Lorentz distribution) because it’s the solution to the differential equation describing forced resonance. In spectroscopy, it is the description of the shape of spectral lines which are subject to homogeneous broadening in which all atoms interact in the same way with the frequency range contained in the line shape.

'''PDF''': 
<math>\frac{1}{\pi\gamma\,\left[1 + \left(\frac{x-x_0}{\gamma}\right)^2\right]}\!</math>

'''CDF''': 
<math>\frac{1}{\pi} \arctan\left(\frac{x-x_0}{\gamma}\right)+\frac{1}{2}\!</math>

'''Mean''': 
Does Not Exist

'''Median''': 
<math>x_0\!</math>

'''Variance''': 
Does Not Exist

'''Support''': 
<math>x \in (-\infty, +\infty)\!</math>

'''1st Moment''': 
None

'''2nd Moment''': 
None

'''Kth Moment''': 
None

===Applications===
<math>\cdot</math> Used in mechanical and electrical theory, physical anthropology and measurement and calibration problems.

<math>\cdot</math> In physics it is called a Lorentzian distribution, where it is the distribution of the energy of an unstable state in quantum mechanics.

<math>\cdot</math> Also used to model the points of impact of a fixed straight line of particles emitted from a point source.

===Example===
[[File:Cauchy.png]]

===SOCR Links===
http://www.distributome.org/ -> SOCR -> Distributions -> Distributome

http://www.distributome.org/ -> SOCR -> Distributions -> Cauchy Distribution

http://www.distributome.org/ -> SOCR -> Functors -> Cauchy Distribution

SOCR Cauchy Distribution calculator (http://socr.ucla.edu/htmls/dist/Cauchy_Distribution.html)

<hr>
* SOCR Home page: http://www.socr.ucla.edu

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	'''Median''': <br>		'''Median''': <br>
		+	<math>x_0\!</math>
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		+	'''Mode''': <br>
	<math>x_0\!</math>		<math>x_0\!</math>

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 <math>x \in (-\infty, +\infty)\!</math>
-'''Moment Generating Function'''
+'''Moment Generating Function''' <br>
 Does Not Exist

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 === Cauchy Distribution ===
-The standard Cauchy distribution is derived from the ratio of two independent Normal distributions. If X ~ N(0,1), and Y ~ N(0,1), then <math>\tfrac{X}{Y} \sim Cauchy(0,1)</math>
+The standard Cauchy distribution is derived from the ratio of two independent [http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Normal_Std Normal Distributions]. If X ~ N(0,1), and Y ~ N(0,1), then <math>\tfrac{X}{Y} \sim Cauchy(0,1)</math>
 The Cauchy distribution is important in physics (where it’s known as the Lorentz distribution) because it’s the solution to the differential equation describing forced resonance. In spectroscopy, it is the description of the shape of spectral lines which are subject to homogeneous broadening in which all atoms interact in the same way with the frequency range contained in the line shape.