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- [section:rayleigh Rayleigh Distribution]
- ``#include <boost/math/distributions/rayleigh.hpp>``
- namespace boost{ namespace math{
- template <class RealType = double,
- class ``__Policy`` = ``__policy_class`` >
- class rayleigh_distribution;
- typedef rayleigh_distribution<> rayleigh;
- template <class RealType, class ``__Policy``>
- class rayleigh_distribution
- {
- public:
- typedef RealType value_type;
- typedef Policy policy_type;
- // Construct:
- rayleigh_distribution(RealType sigma = 1)
- // Accessors:
- RealType sigma()const;
- };
- }} // namespaces
- The [@http://en.wikipedia.org/wiki/Rayleigh_distribution Rayleigh distribution]
- is a continuous distribution with the
- [@http://en.wikipedia.org/wiki/Probability_density_function probability density function]:
- [expression f(x; sigma) = x * exp(-x[super 2]/2 [sigma][super 2]) / [sigma][super 2]]
- For sigma parameter ['[sigma]] > 0, and /x/ > 0.
- The Rayleigh distribution is often used where two orthogonal components
- have an absolute value,
- for example, wind velocity and direction may be combined to yield a wind speed,
- or real and imaginary components may have absolute values that are Rayleigh distributed.
- The following graph illustrates how the Probability density Function(pdf) varies with the shape parameter [sigma]:
- [graph rayleigh_pdf]
- and the Cumulative Distribution Function (cdf)
- [graph rayleigh_cdf]
- [h4 Related distributions]
- The absolute value of two independent normal distributions X and Y, [radic] (X[super 2] + Y[super 2])
- is a Rayleigh distribution.
- The [@http://en.wikipedia.org/wiki/Chi_distribution Chi],
- [@http://en.wikipedia.org/wiki/Rice_distribution Rice]
- and [@http://en.wikipedia.org/wiki/Weibull_distribution Weibull] distributions are generalizations of the
- [@http://en.wikipedia.org/wiki/Rayleigh_distribution Rayleigh distribution].
- [h4 Member Functions]
- rayleigh_distribution(RealType sigma = 1);
- Constructs a [@http://en.wikipedia.org/wiki/Rayleigh_distribution
- Rayleigh distribution] with [sigma] /sigma/.
- Requires that the [sigma] parameter is greater than zero,
- otherwise calls __domain_error.
- RealType sigma()const;
- Returns the /sigma/ parameter of this distribution.
- [h4 Non-member Accessors]
- All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all
- distributions are supported: __usual_accessors.
- The domain of the random variable is \[0, max_value\].
- [h4 Accuracy]
- The Rayleigh distribution is implemented in terms of the
- standard library `sqrt` and `exp` and as such should have very low error rates.
- Some constants such as skewness and kurtosis were calculated using
- NTL RR type with 150-bit accuracy, about 50 decimal digits.
- [h4 Implementation]
- In the following table [sigma] is the sigma parameter of the distribution,
- /x/ is the random variate, /p/ is the probability and /q = 1-p/.
- [table
- [[Function][Implementation Notes]]
- [[pdf][Using the relation: pdf = x * exp(-x[super 2])/2 [sigma][super 2] ]]
- [[cdf][Using the relation: p = 1 - exp(-x[super 2]/2) [sigma][super 2]= -__expm1(-x[super 2]/2) [sigma][super 2]]]
- [[cdf complement][Using the relation: q = exp(-x[super 2]/ 2) * [sigma][super 2] ]]
- [[quantile][Using the relation: x = sqrt(-2 * [sigma] [super 2]) * log(1 - p)) = sqrt(-2 * [sigma] [super 2]) * __log1p(-p))]]
- [[quantile from the complement][Using the relation: x = sqrt(-2 * [sigma] [super 2]) * log(q)) ]]
- [[mean][[sigma] * sqrt([pi]/2) ]]
- [[variance][[sigma][super 2] * (4 - [pi]/2) ]]
- [[mode][[sigma] ]]
- [[skewness][Constant from [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
- [[kurtosis][Constant from [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
- [[kurtosis excess][Constant from [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Weibull Distribution." From MathWorld--A Wolfram Web Resource.] ]]
- ]
- [h4 References]
- * [@http://en.wikipedia.org/wiki/Rayleigh_distribution ]
- * [@http://mathworld.wolfram.com/RayleighDistribution.html Weisstein, Eric W. "Rayleigh Distribution." From MathWorld--A Wolfram Web Resource.]
- [endsect] [/section:Rayleigh Rayleigh]
- [/
- Copyright 2006 John Maddock and Paul A. Bristow.
- Distributed under the Boost Software License, Version 1.0.
- (See accompanying file LICENSE_1_0.txt or copy at
- http://www.boost.org/LICENSE_1_0.txt).
- ]
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