## What is the Rayleigh Distribution?

The Rayleigh distribution is a continuous probability distribution named after the English Lord Rayleigh. It 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:

Where *e* is Euler’s number.

As the shape parameter increases, the distribution gets wider and flatter.

The distribution is widely used:

- 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.

## Variance and Mean (Expected Value) of a Rayleigh Distribution

The expected value (the mean) of a Rayleigh 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 the improper integral:

Where:

*exp*is the exponential function,- dx is the differential operator.

Solving the integral for you gives the Rayleigh expected value of σ √(π/2)

The variance of a Rayleigh distribution is derived in a similar way, giving the variance formula of:

Var(x) = σ^{2}((4 – π)/2).

## References

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*A 3-Component Mixture: Properties and Estimation in Bayesian Framework.* Aslam et. al. Retrieved October 3, 2015. Available here.