Rayleigh Distribution

The Rayleigh distribution, also known as the Rician distribution, is used to model the magnitude of a vector whose components are independent and normally distributed.

The Rayleigh distribution has a scale parameter $\sigma$. The probability density function (PDF) is given by:

$$f(x;\sigma )=\frac{x}{{\sigma }^2}{e}^{-\frac{x^2}{2{\sigma }^2}}\text{for }x\ge 0$$

The cumulative distribution function (CDF) is:

$$F(x;\sigma )=1-e^{-\frac{x^2}{2{\sigma }^2}}\text{for }x\ge 0$$

The Rayleigh distribution is defined for all non-negative real numbers ($x\ge 0$), with its scale parameter $\sigma$ required to be strictly positive ($\sigma >0$).

• Wind speed modeling uses the Rayleigh distribution in meteorology.

• Signal processing employs it for modeling noise in communication systems.

• Wave height analysis in oceanography utilizes the distribution.

• Random walk models in physics use it for distance calculations.

The Rayleigh distribution arises naturally as the distribution of the magnitude of a two-dimensional vector with independent, normally distributed components.

Notable properties include:

• The characteristic function is $\varphi (t)=1+\sigma t{e}^{-{\sigma }^2{t}^2/4}(\sqrt{\pi }/2)(1-\mathrm{erf}(\sigma t/2))$.

• The hazard rate is monotonically increasing.

• The distribution has a single mode at $\sigma$.

• If X and Y are independent normal variables with mean 0 and variance ${\sigma }^2$, then $\sqrt{X^2+Y^2}$ follows a Rayleigh distribution.

• The Rayleigh distribution is a special case of the chi distribution with 2 degrees of freedom.

• It is related to the Maxwell distribution in three dimensions.

The Rayleigh distribution is implemented by the RayleighDistribution class. It has one constructor that takes the scale parameter as its only argument.

The following constructs the Rayleigh distribution with scale parameter 1.8:

var rayleigh = new RayleighDistribution(1.8);

Dim rayleigh = New RayleighDistribution(1.8)

No code example is currently available or this language may not be supported.

let rayleigh = RayleighDistribution(1.8)

The RayleighDistribution class has two specific properties, LocationParameter and ScaleParameter, which return the location parameter (smallest possible value) and scale parameter of the distribution.

RayleighDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. The second and third parameters are the location and scale parameters of the distribution.

var random = new Pcg32();
double sample = RayleighDistribution.Sample(random, 1.8);

Dim random = New Pcg32()
Dim sample = RayleighDistribution.Sample(random, 1.8)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = RayleighDistribution.Sample(random, 1.8)

The above example uses the MersenneTwister class to generate uniform random numbers.

For more information on the Rayleigh distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 18. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 39. Wiley Series in Probability and Statistics.

• "Probability and Statistics for Engineers and Scientists" by Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, and Keying E. Ye

• "Introduction to Probability Models" by Sheldon M. Ross

Other Resources