Calculate PDF, CDF, quantiles, and moments for the Rayleigh distribution. Includes distribution visualization, survival function, hazard rate, and range probabilities.
The Rayleigh Distribution Calculator computes the probability density function (PDF), cumulative distribution function (CDF), survival function, hazard rate, quantiles, and moments for the Rayleigh distribution. Simply enter the scale parameter σ and evaluate at any point x.
The Rayleigh distribution models the magnitude of a 2D vector whose components are independent, zero-mean Gaussian random variables with equal variance σ². It appears naturally in wind speed modeling, signal processing (envelope of a narrowband signal), communications engineering, and physics (particle velocity magnitudes in a 2D gas).
One unique property of the Rayleigh distribution is its linearly increasing hazard function h(x) = x/σ², meaning the failure rate grows proportionally with time or distance. This makes it suitable for modeling wear-out failures in reliability engineering and the aging of certain components. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results.
The Rayleigh distribution appears across multiple engineering and scientific disciplines, yet computing its properties by hand involves exponential functions and special constants. This calculator provides instant PDF, CDF, quantile, and moment calculations with visual aids.
Engineers modeling wind speeds, signal processing researchers computing envelope statistics, and reliability analysts studying wear-out failures all benefit from having a comprehensive Rayleigh distribution tool with visualization and lookup tables.
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.
Common applications include wind speed modeling, sound signal amplitude (envelopes), wave heights in oceanography, lifetime analysis in reliability engineering, and modeling distances in 2D random walks. Use this as a practical reminder before finalizing the result.
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.