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The Rayleigh distribution is a continuous probability distribution that models the magnitude of a random vector.

The Rayleigh distribution is often used in engineering and physics to model the magnitude of a random vector, such as the wind speed or the amplitude of an electromagnetic wave. It is a continuous probability distribution with a probability density function given by:

f(x; σ) = (x/σ^2) * exp(-x^2/(2σ^2))

where x is the magnitude of the random vector, σ is a scale parameter that determines the spread of the distribution, and exp is the exponential function.

The mean and variance of the Rayleigh distribution are given by:

μ = σ * sqrt(π/2)

σ^2 = (4 - π)/2 * σ^2

The mode of the distribution is equal to σ, and the median is equal to σ * sqrt(ln(2)). The cumulative distribution function of the Rayleigh distribution is given by:

F(x; σ) = 1 - exp(-x^2/(2σ^2))

The Rayleigh distribution is a special case of the Weibull distribution, with the shape parameter equal to 2. It is also related to the chi distribution, with the square of a random variable following a chi distribution with two degrees of freedom being Rayleigh-distributed.

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