Rayleigh Distribution Calculator — PDF, CDF, Mean & Variance
Enter the scale parameter σ and an evaluation point x to compute the Rayleigh PDF and CDF, plus the distribution mean, mode, median and variance.
Probability that X does not exceed 1.5
- 1
x²
1.5² = 2.25 - 2
Exponent argument (x² ÷ 2σ²)
2.25 ÷ (2 × 1²) = 1.125 - 3
exp(−argument)
exp(−1.125) = 0.324652The survival function of the squared-exponential term. - 4
CDF = 1 − exp(−x²/2σ²)
1 − 0.324652 = 0.675348
How does this calculator work?
The Rayleigh distribution with scale σ has PDF f(x) = (x/σ²)·exp(−x²/(2σ²)) and CDF F(x) = 1 − exp(−x²/(2σ²)) for x ≥ 0. Its mean is σ√(π/2), mode is σ, and variance is (4−π)/2·σ². Used in wind speed modelling, wireless fading, and ocean wave heights.
Formula
How this is calculated
The Rayleigh distribution models the magnitude (amplitude) of a 2-D vector whose two orthogonal components are each independent zero-mean Gaussian random variables with the same standard deviation σ. In physical terms, if you measure a wind vector with random east and north components both having zero mean and SD = σ, the wind speed (the resultant magnitude) follows a Rayleigh distribution with scale σ.
The PDF, f(x; σ) = (x / σ²) · exp(−x² / (2σ²)), rises from zero at x = 0, peaks at the mode x = σ, and decays to zero as x → ∞. Unlike the normal distribution it is defined only for x ≥ 0. The CDF, F(x; σ) = 1 − exp(−x² / (2σ²)), gives the probability that a random observation does not exceed x — useful for exceedance probability in reliability and signal processing.
Key distributional properties: mean = σ√(π/2) ≈ 1.253σ; median = σ√(2 ln 2) ≈ 1.177σ; mode = σ; variance = (4 − π)/2 · σ² ≈ 0.429σ². Applications include modelling wind speeds, ocean wave heights, signal envelopes in communications (e.g. Rayleigh fading), and lifetime data in reliability analysis.
Frequently asked questions
The square of a Rayleigh-distributed random variable follows a chi-squared distribution with 2 degrees of freedom (scaled by σ²). Equivalently, if X ~ Rayleigh(σ), then X² ~ Exponential(1/(2σ²)), which is a chi-squared(2)/2 distribution.
In wind energy assessment the Rayleigh distribution is often used as a one-parameter approximation to the Weibull distribution (setting the shape parameter k = 2). Given a mean wind speed v̄, the scale parameter is σ = v̄ / √(π/2). The CDF then gives the fraction of time wind speed is below any threshold.
When a radio signal reaches a receiver via many scattered paths with no dominant line-of-sight component, the signal envelope (amplitude) follows a Rayleigh distribution. This Rayleigh fading model is a standard baseline in wireless channel modelling.
Also known as
TG we-Calculate Editorial Team. (2026). Rayleigh Distribution Calculator — PDF, CDF, Mean & Variance [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/rayleigh-distribution-calculator
TG we-Calculate Editorial Team. "Rayleigh Distribution Calculator — PDF, CDF, Mean & Variance." TG we-Calculate. 2026. https://we-calculate.com/calculator/rayleigh-distribution-calculator.
TG we-Calculate Editorial Team, "Rayleigh Distribution Calculator — PDF, CDF, Mean & Variance," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/rayleigh-distribution-calculator
@misc{wecalculate_rayleigh_distribution_calculator, title = {Rayleigh Distribution Calculator — PDF, CDF, Mean & Variance}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/rayleigh-distribution-calculator}}, year = {2026}, note = {TG we-Calculate} }
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