Difference between revisions of "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. | ||
| − | '''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> | ||
| − | '''CDF''' <br> | + | '''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> | ||
| − | '''Mean''' <br> | + | '''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> | ||
| − | '''Variance''' <br> | + | '''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> | ||
| − | '''Support''' <br> | + | '''Support''': <br> |
''x'' ∈ [0, +∞) | ''x'' ∈ [0, +∞) | ||
===Moments=== | ===Moments=== | ||
| − | The first few [ | + | 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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Latest revision as of 12:09, 3 March 2020
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
- SOCR Home page: http://www.socr.ucla.edu
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