Rayleigh(b) - no Crystal Ball distribution = Weibull(b√2,2)

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.

#### Uses

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.

The distance from one individual to its nearest neighbor 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/b^{2}.

#### Construction

The Rayleigh distribution can be modeled with the use of the Weibull distribution using the following:

Rayleigh(b) = Weibull(2, b√2),

#### Comments

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