Rayleigh distribution

Format: Rayleigh(b)


Originally derived by Lord Rayleigh (or by his less glamorous name J.W. Strutt) in the field of acoustics.

The graph below shows various Rayleigh distributions. The distribution in black is a Rayleigh(1), sometimes referred to as the standard Rayleigh distribution.


The Rayleigh distribution is frequently used to model wave heights in oceanography, and in communication theory to describe hourly median and instantaneous peak power of received radio signals. It has been used to model the frequency of different wind speeds over a year at wind turbine sites.

The distance from one individual to its nearest neighbour when the spatial pattern is generated by a Poisson distribution follows a Rayleigh distribution. This example shows how that turns out to be very useful.

Consider the location of an object in two dimensions {x,y} relative to some point at location {0,0}. Imagine that x = Normal(0,s) and y = x = Normal(0,s), where the two distributions are independent. Then the distance of the object from point {0,0} is given by a Rayleigh(s) distribution. In other words, SQRT( Normal(0,s)^2 + Normal(0,s)^2 ) = Rayleigh(s)

The Rayleigh distribution is a special case of the Weibull distribution since Rayleigh(b) = Weibull(2, b√2), and as such is a suitable distribution for modeling the lifetime of a device that has a linearly increasing instantaneous failure rate: z(x) = x/b2.  



Other identities: [Rayleigh (1)]2 = ChiSq (2) and [Rayleigh(β)]2 = Expon(1/(2β2)).

ModelRisk functions added to Microsoft Excel for the Rayleigh distribution

VoseRayleigh generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseRayleighObject constructs a distribution object for this distribution.

VoseRayleighProb returns the probability density or cumulative distribution function for this distribution.

VoseRayleighProb10 returns the log10 of the probability density or cumulative distribution function.  

VoseRayleighFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseRayleighFitObject constructs a distribution object of this distribution fitted to data.

VoseRayleighFitP returns the parameters of this distribution fitted to data.


Rayleigh distribution equations






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