Rayleigh distribution calculator

The formulas

f(x) = (x / σ²) e^{−x² / (2σ²)}, x ≥ 0 F(x) = 1 − e^{−x² / (2σ²)} x_p = σ √(−2 ln(1 − p))

mean = σ√(π/2) variance = (4 − π)σ² / 2 median = σ√(2 ln 2) mode = σ

Worked example

The horizontal miss distance of a drone's landing from its target has east and north errors that are each normal with mean 0 and σ = 2 metres. What is the probability that it lands within 3 metres of the target?

P(X ≤ 3) = 1 − e^{−3² / (2 × 2²)} = 1 − e^−9/8 = 1 − 0.324652 = 0.675348

About 67.5% of landings are within 3 m, and 32.5% are further out. The density at 3 m is (3/4) × 0.324652 = 0.243489 per metre.

The most likely miss distance is the mode, σ = 2 m. The median is 2√(2 ln 2) = 2.35482 m, and the mean is 2√(π/2) = 2.50663 m, with variance (4 − π) × 4 / 2 = 1.71681 and standard deviation 1.31027 m. To find the radius that contains 90% of landings, use the percentile formula: 2 × √(−2 ln 0.1) = 4.29193 m. The 50% radius, the median 2.35482 m, is what ballistics calls the circular error probable (CEP).

Reading the result

The curve starts at zero, rises to its peak at σ and has a moderate right tail. It starts at zero because hitting very close to the centre needs both components to be small at once, which is unlikely even though each component alone is most likely near 0. The shaded area on the chart is the probability you asked for.

Where it is used

Signal engineers use the Rayleigh distribution for the envelope of a radio signal with no direct line of sight, where many reflected paths add up with random phases. Wind-energy studies use it as a one-parameter wind-speed model when only the average speed is known: set σ = mean / 1.2533. Ocean engineers use it for individual wave heights in a sea state. In each case the physical picture is the same, a length built from two independent, equally spread normal components.

Common mistakes

• Entering the SD of the distance as σ. σ is the SD of each component (and the mode); the SD of the distance is only 0.655σ.
• Using the Rayleigh when the components have different spreads or a non-zero mean. Then the length follows a Hoyt or Rice distribution instead.
• Treating the density as a probability. f(3) = 0.2435 is a height per metre, not the chance of exactly 3 m.