# AP Statistics Curriculum 2007 Rice

(Difference between revisions)
 Revision as of 04:28, 7 July 2011 (view source)JayZzz (Talk | contribs)← Older edit Revision as of 04:32, 7 July 2011 (view source)JayZzz (Talk | contribs) Newer edit → Line 4: Line 4: 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'''
+ '''PDF''':
$\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)} [itex]\frac{x}{\sigma^2}\exp\left(\frac{-(x^2+\nu^2)} {2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)$ {2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right)[/itex] - '''CDF'''
+ '''CDF''':
$1-Q_1\left(\frac{\nu}{\sigma },\frac{x}{\sigma }\right)$ $1-Q_1\left(\frac{\nu}{\sigma },\frac{x}{\sigma }\right)$ - '''Mean'''
+ '''Mean''':
$\sigma \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)$ $\sigma \sqrt{\pi/2}\,\,L_{1/2}(-\nu^2/2\sigma^2)$ - '''Variance'''
+ '''Variance''':
$2\sigma^2+\nu^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)$ $2\sigma^2+\nu^2-\frac{\pi\sigma^2}{2}L_{1/2}^2\left(\frac{-\nu^2}{2\sigma^2}\right)$ - '''Support'''
+ '''Support''':
''x'' ∈ [0, +∞) ''x'' ∈ [0, +∞)

## 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

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

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

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