Calculate PDF, CDF, quantiles, and moments for the Rayleigh distribution. Includes distribution visualization, survival function, hazard rate, and range probabilities.
The Rayleigh Distribution Calculator evaluates the PDF, CDF, survival function, hazard rate, quantiles, and moments for a Rayleigh distribution.
The distribution appears when you measure the magnitude of a two-dimensional vector whose components are independent zero-mean Gaussian variables. That shows up in wind-speed modeling, signal envelopes, communications engineering, and other settings where a length or amplitude is built from orthogonal noise components.
Because the hazard function increases linearly, the page also helps with wear-out style reliability problems where the instantaneous failure rate rises as the process ages.
Rayleigh calculations are repetitive by hand because the same scale parameter drives the PDF, CDF, moments, and quantiles. Having the curve, the summary values, and the threshold probabilities together makes it easier to interpret the shape instead of treating each value in isolation.
PDF: f(x) = (x/σ²) exp(-x²/2σ²) for x ≥ 0. CDF: F(x) = 1 - exp(-x²/2σ²). Mean: σ√(π/2). Median: σ√(2 ln 2). Mode: σ. Variance: (4-π)/2 × σ².
Result: f(15) = 0.01648, F(15) = 0.6753, Mean = 12.533
With σ = 10: f(15) = (15/100) × exp(-225/200) = 0.01648. CDF = 1 - exp(-225/200) = 0.6753. Mean = 10√(π/2) ≈ 12.533. About 67.5% of values fall below x = 15.
Lord Rayleigh first described this distribution in 1880 when studying the problem of adding together many harmonic vibrations with random phases — the amplitude of the resultant sum follows a Rayleigh distribution. Today it appears whenever we compute the magnitude of a 2D Gaussian vector, from GPS error to ocean wave heights.
The Rayleigh distribution belongs to a family of related distributions. It is a special case of the Weibull distribution (shape k=2), the chi distribution (2 df), and the Rice distribution (ν=0). When the underlying Gaussian components have non-zero means, the magnitude follows a Rice distribution instead.
In wind energy engineering, the Rayleigh distribution is the standard model for wind speed at a given location. The mean wind speed determines σ via the relation σ = v̄/√(π/2). Turbine designers use the CDF to estimate the fraction of time wind speed exceeds the cut-in speed and the fraction below the rated speed, directly informing capacity factor calculations.
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Common applications include wind-speed modeling, signal envelopes, wave heights, and reliability problems where the hazard rate rises with size or age.
σ sets the spread of the distribution. The mode (peak) equals σ, the mean is σ√(π/2) ≈ 1.253σ, and the median is σ√(2 ln 2) ≈ 1.177σ. Larger σ spreads the distribution rightward.
If X and Y are independent N(0, σ²) variables, then R = √(X²+Y²) follows a Rayleigh(σ) distribution. It's the distance from the origin in 2D Gaussian noise.
The hazard function h(x) = x/σ² is linearly increasing, meaning the instantaneous failure rate grows proportionally with x. This is characteristic of wear-out failure modes in reliability analysis.
No, it is right-skewed with a fixed skewness of about 0.631. It only takes non-negative values (x ≥ 0) and has a single peak at x = σ.
The Rayleigh distribution is a special case of the chi distribution with 2 degrees of freedom. It is also related to the Weibull distribution with shape parameter k = 2.