How this calculator works
Rayleigh values are non-negative magnitudes. Scale is the SD of each underlying normal component, rather than the SD of the magnitude. Changing scale stretches the x-axis. The squared magnitude divided by twice the scale squared has a unit-rate exponential distribution.
Formula
f(x) = (x/σ²)exp(-x²/(2σ²)); F(x) = 1-exp(-x²/(2σ²)); E[X] = σ√(π/2); Var[X] = (2-π/2)σ².
Worked example
For scale 2 and x = 2, cumulative probability is 1-exp(-0.5) = 0.3934693 and density is exp(-0.5)/2 = 0.3032653.
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Frequently asked questions
Is the scale the mean?
No. Multiply scale by sqrt(π/2) to obtain the mean.
Can I use a nonzero component mean?
That generally leads to a Rice distribution rather than Rayleigh.
Why are negative values impossible?
The variable represents a magnitude, which cannot be negative.