#include <boost/math/distributions/rayleigh.hpp>
namespace boost{ namespace math{ template <class RealType = double, class Policy = policies::policy<> > 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 Rayleigh distribution is a continuous distribution with the probability density function:
f(x; sigma) = x * exp(x^{2}/2 σ^{2}) / σ^{2}
For sigma parameter σ > 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 σ:
and the Cumulative Distribution Function (cdf)
The absolute value of two independent normal distributions X and Y, √ (X^{2} + Y^{2}) is a Rayleigh distribution.
The Chi, Rice and Weibull distributions are generalizations of the Rayleigh distribution.
rayleigh_distribution(RealType sigma = 1);
Constructs a Rayleigh distribution with σ sigma.
Requires that the σ parameter is greater than zero, otherwise calls domain_error.
RealType sigma()const;
Returns the sigma parameter of this distribution.
All the usual nonmember accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.
The domain of the random variable is [0, max_value].
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 150bit accuracy, about 50 decimal digits.
In the following table σ is the sigma parameter of the distribution, x is the random variate, p is the probability and q = 1p.
Function 
Implementation Notes 


Using the relation: pdf = x * exp(x^{2})/2 σ^{2} 
cdf 
Using the relation: p = 1  exp(x^{2}/2) σ^{2}= expm1(x^{2}/2) σ^{2} 
cdf complement 
Using the relation: q = exp(x^{2}/ 2) * σ^{2} 
quantile 
Using the relation: x = sqrt(2 * σ ^{2}) * log(1  p)) = sqrt(2 * σ ^{2}) * log1p(p)) 
quantile from the complement 
Using the relation: x = sqrt(2 * σ ^{2}) * log(q)) 
mean 
σ * sqrt(π/2) 
variance 
σ^{2} * (4  π/2) 
mode 
σ 
skewness 
Constant from Weisstein, Eric W. "Weibull Distribution." From MathWorldA Wolfram Web Resource. 
kurtosis 
Constant from Weisstein, Eric W. "Weibull Distribution." From MathWorldA Wolfram Web Resource. 
kurtosis excess 
Constant from Weisstein, Eric W. "Weibull Distribution." From MathWorldA Wolfram Web Resource. 