Rayleigh Distribution Calculator & Guide

The Rayleigh Distribution. Why can distance from a target have a most likely value above zero when each coordinate error is centered at zero? Take two independent centered normal coordinates with common spread sigma, then measure their vector length. That nonnegative length is Rayleigh. Sigma must be positive, and it is both the component spread and the magnitude mode. Its density is x over sigma squared times exponential negative x squared over two sigma squared. The cumulative probability is one minus that exponential, so the survival tail is the exponential term itself. Invert the cumulative function to get quantile p as sigma times square root of negative two log one minus p.

Expanding circles create more possible directions while normal decay removes distant points, producing one interior peak. The mode is sigma, the median is about one point one seven seven sigma, and the mean is about one point two five three sigma. Variance is four minus pi over two times sigma squared, so magnitude standard deviation is only about point six five five sigma. Every positive scale has the same mild right skew, about point six three one, and excess kurtosis about point two four five. Beyond k sigma, probability is exponential negative k squared over two: about point one three five beyond two sigma. Square the magnitude and the model becomes exponential with mean two sigma squared, a powerful calculation and diagnostic identity. Rayleigh is also Weibull shape two, but Weibull scale is sigma times square root of two, not sigma itself. Use it for no-line-of-sight signal envelopes, isotropic radial errors, or wind magnitude when the component assumptions are plausible. A nonzero mean vector produces Rice behavior, while unequal or correlated component spreads require a more general family.

Suppose north and east landing errors are independent centered normals, each with sigma ten meters. The median miss, also called circular error probable here, is ten square root two log two, or eleven point seven seven meters. Probability within fifteen meters is one minus exponential negative one point one two five, approximately point six seven five three. The ninety-fifth percentile radius is ten square root negative two log point zero five, approximately twenty-four point four eight meters. As an audit, squared miss distance has exponential mean two hundred square meters, giving the same thirty-two point five percent beyond fifteen meters. One component spread now yields a median radius, coverage probability, and operational ninety-five percent radius with consistent units.

With data, square the observed magnitudes and check whether the transformed values look exponential. Also verify that the density peak occurs at x equal sigma and that no observations are negative. Reject nonpositive sigma, nonfinite inputs, and quantile probabilities outside the open unit interval. Do not use a magnitude model for signed one-dimensional measurements or infer it from a right-skewed histogram alone. A visible deterministic signal or directional offset is evidence for Rice rather than Rayleigh. Report sigma, the two-component assumptions, event boundary, units, population, and whether results describe radius or squared radius. Explore Rayleigh probabilities, quantiles, vector geometry, and squared-magnitude checks free at Distri Scope dot com.