The Rayleigh Distribution

The Rayleigh distribution density function has the following form:

This distribution shows up in the aforementioned case when converting two independent normal distributions with the same variance from rectangular to polar coordinates. This a a very common situation because it is the case of looking for the radius distance from the origin in a polar coordinate system. Just as the angle distribution in polar coodinates are uniformly distributed (see Section 11.6), the radial distance from the origin is Rayleigh-distributed. Thus, for the bull's-eye example, the vector from the origin obeys the Ray-leigh law under the circumstances described before.

Unlike the normal distribution, the Rayleigh can be integrated in a straightforward manner. Thus, the CDF for the Rayleigh becomes

and represents, as usual, the probability that the variable is less than or equal to a particular value of x.

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