AP Statistics Curriculum 2007 Rice

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Also known as the Rician Distribution, the Rice distribution is the probability distribution of the absolute value of a Circular Bivariate Normal random variable with potentially non-zero mean.  
Also known as the Rician Distribution, the Rice distribution is the probability distribution of the absolute value of a Circular Bivariate Normal random variable with potentially non-zero mean.  
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'''PDF''' <br>
+
'''PDF''': <br>
<math>\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)}
<math>\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)}
{2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)</math>
{2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)</math>
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'''CDF''' <br>
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'''CDF''': <br>
<math>1-Q_1\left(\frac{\nu}{\sigma },\frac{x}{\sigma }\right)</math>
<math>1-Q_1\left(\frac{\nu}{\sigma },\frac{x}{\sigma }\right)</math>
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'''Mean''' <br>
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'''Mean''': <br>
<math>\sigma  \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)</math>
<math>\sigma  \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)</math>
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'''Variance''' <br>
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'''Variance''': <br>
<math>2\sigma^2+\nu^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)</math>
<math>2\sigma^2+\nu^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)</math>
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'''Support''' <br>
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'''Support''': <br>
''x'' ∈ [0,&thinsp;+∞)
''x'' ∈ [0,&thinsp;+∞)
===Moments===
===Moments===
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The first few [[raw moments]] are:
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The first few [http://en.wikipedia.org/wiki/Raw_moment raw moments] are:
:<math>\mu_1^'=  \sigma  \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)</math>
:<math>\mu_1^'=  \sigma  \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)</math>
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* SOCR Home page: http://www.socr.ucla.edu
* 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_Johnson_SB}}
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{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Rice}}

Current revision as of 22:44, 18 July 2011

Contents

General Advance-Placement (AP) Statistics Curriculum - Rice Distribution

Rice Distribution

Also known as the Rician Distribution, the Rice distribution is the probability distribution of the absolute value of a Circular Bivariate Normal random variable with potentially non-zero mean.

PDF:
\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)}
{2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)

CDF:
1-Q_1\left(\frac{\nu}{\sigma },\frac{x}{\sigma }\right)

Mean:
\sigma  \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)

Variance:
2\sigma^2+\nu^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)

Support:
x ∈ [0, +∞)

Moments

The first few raw moments are:

\mu_1^'=  \sigma  \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)
\mu_2^'= 2\sigma^2+\nu^2\,

Applications

Describing fading in wireless communications systems analysis
http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1532472

Radar signal processing
http://users.ece.gatech.edu/mrichard/Rice%20power%20pdf.pdf

Describing noisy MRI Data
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2254141/

SOCR Links

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

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

http://www.distributome.org/ -> SOCR -> Modeler -> Rice_Fit Modeler

SOCR Rice Distribution Calculator: http://socr.ucla.edu/htmls/dist/Rice_Distribution.html




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