The Rayleigh distribution describes the length of a two-dimensional vector whose two components are independent normal variables with mean 0 and the same standard deviation σ: R = √(X² + Y²). Whenever a magnitude is built from two random, perpendicular, equally variable parts, the Rayleigh distribution is a natural model.
It is used for wind speed (east–west and north–south components), the heights of ocean waves, the amplitude of a radio signal with scattered paths (Rayleigh fading), the miss distance of shots around a target and the noise envelope in radar and communications. It is the Weibull distribution with shape k = 2, and the square of a Rayleigh variable is exponential.
The single parameter σ is also the mode – the most likely value. The calculator gives the probability below, above, between or outside chosen values, quantiles from the closed-form inverse, random values and all the usual summary measures. A larger σ stretches the curve to the right and flattens it, but the shape – a mild right skew – never changes.
How to use the Rayleigh Distribution Calculator
- Scale parameter σ
- What to calculate
- Value x ≥ 0 (x₁ for a range)
- Upper value x₂
- Cumulative probability p
- How many random values
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
All probabilities use the closed-form CDF 1 − e^(−x²/(2σ²)); quantiles invert it exactly as σ√(−2 ln(1 − p)), which also generates random values. Choose what to calculate: a cumulative probability, an upper tail, the probability between or outside two values, a quantile (inverse CDF) or a random sample generated by inverse-transform sampling. The chart shades exactly the area you asked for.
Worked example
With σ = 2 the probability of a value up to 3 is 1 − e^(−3²/(2 × 4)) = 1 − e^(−1.125) = 67.53%. The mean is 2√(π/2) = 2.5066, the median 2√(2 ln 2) = 2.3548, the mode 2, the variance (4 − π)/2 × 4 = 1.7168 and the 90th percentile 2√(−2 ln 0.1) = 4.2919.
Wind example: if hourly wind speed follows a Rayleigh distribution with σ = 6 m/s, the chance of more than 15 m/s is e^(−225/72) = 4.39%.
Summary measures
| Measure | Formula | Value for σ = 1 |
|---|---|---|
| Mean | σ√(π/2) | 1.2533 |
| Median | σ√(2 ln 2) | 1.1774 |
| Mode | σ | 1 |
| Variance | (4 − π)σ²/2 | 0.4292 |
| Skewness | 2√π(π − 3)/(4 − π)^1.5 | 0.6311 |
| Excess kurtosis | −(6π² − 24π + 16)/(4 − π)² | 0.2451 |
Formula
f(x) = (x/σ²) e^(−x²/(2σ²)) · P(X ≤ x) = 1 − e^(−x²/(2σ²)) · quantile = σ√(−2 ln(1 − p)) · mean = σ√(π/2) · median = σ√(2 ln 2) · mode = σ · variance = (4 − π)σ²/2
Worked example
Inputs — Scale parameter σ: 2; What to calculate: P(X ≤ x) – probability at or below x; Value x ≥ 0 (x₁ for a range): 3.
Results — Result = P(X ≤ 3) = 0.675348 (67.5348%); Probability density f(x) = 0.243489; P(X ≤ x) – cumulative (CDF) = 67.5348 %; P(X > x) – survival = 32.4652 %; Mean (expected value) = 2.5066; Median = 2.3548.
Key terms
- Scale parameter (σ)
- The standard deviation of each normal component; also the mode of the Rayleigh distribution.
- Magnitude
- The length √(X² + Y²) of a vector with components X and Y.
- Rayleigh fading
- Random variation of a radio signal's strength when there is no direct line of sight.
Frequently asked questions
What is the Rayleigh distribution?
The distribution of √(X² + Y²) when X and Y are independent N(0, σ²) – the length of a random 2-D vector.
What is P(X ≤ 3) for σ = 2?
1 − e^(−9/8) = 67.53%.
What are the mean and mode?
The mode is σ and the mean σ√(π/2) ≈ 1.2533σ; for σ = 2 they are 2 and 2.5066.
How is it related to the Weibull distribution?
It is a Weibull distribution with shape k = 2 and scale λ = σ√2.
Where is it used?
Wind speeds, significant wave heights, Rayleigh fading in wireless channels, radar noise, and the radial error of shots or GPS fixes.
How do I estimate σ from data?
The maximum-likelihood estimate is σ̂ = √(Σxᵢ² / (2n)).