What is the Rayleigh Distribution?
- In communications theory, to model multiple paths of dense scattered signals reaching a receiver.
- In the physical sciences to model wind speed, wave heights and sound/light radiation.
- In engineering, to measure the lifetime of an object, where the lifetime depends on the object’s age. For example: resistors, transformers, and capacitors in aircraft radar sets.1
- In medical imaging science, to model noise variance in magnetic resonance imaging.
The Rayleigh distribution is a special case of the Weibull distribution with a scale parameter of 2. When a Rayleigh is set with a shape parameter (σ) of 1, it is equal to a chi square distribution with 2 degrees of freedom.
The notation X Rayleigh(σ) means that the random variable X has a Rayleigh distribution with shape parameter σ. The probability density function (X > 0) is:
As the shape parameter increases, the distribution gets wider.
Variance and Mean (Expected Value) of a Rayleigh Distribution
The expected value (the mean) of a Rayleigh distribution is:
How this equation is derived involves solving an integral, using calculus:
The expected value of a probability distribution is:
E(x) = ∫ xf(x)dx.
Substituting in the Rayleigh probability density function, this becomes:
This Wolfram calculator will solve the integral for you, giving the Rayleigh expected value of σ √(π/2)
The variance of a Rayleigh dist is derived in a similar way, giving the variance formula of:
Var(x) = σ2((4 – π)/2).
A 3-Component Mixture: Properties and Estimation in Bayesian Framework. Aslam et. al. Retrieved October 3, 2015. Available here.
Stephanie Glen. "Rayleigh Distribution: Definition, Uses, Mean, Variance" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/rayleigh-distribution/
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