Rayleigh Distribution Calculator
Enter the scale parameter σ and x value to explore Rayleigh distribution probabilities and metrics.
PDF f(x): 0.243489
CDF F(x): 0.675348
Survival S(x): 0.324652
Mean: 2.5066
Variance: 1.7168
Mode: 2.0000
Density samples
| x | f(x) |
|---|---|
| 0.00 | 0.000000 |
| 0.80 | 0.184623 |
| 1.60 | 0.290460 |
| 2.40 | 0.292051 |
| 3.20 | 0.222430 |
| 4.00 | 0.135335 |
| 4.80 | 0.067362 |
| 5.60 | 0.027778 |
| 6.40 | 0.009562 |
| 7.20 | 0.002761 |
| 8.00 | 0.000671 |
How to Use This Calculator
- Provide the Rayleigh scale parameter σ (must be positive).
- Enter an x value to evaluate density and cumulative probability.
- Review survival probability, mean, variance, and mode.
- Use the density samples to visualize the Rayleigh curve if needed.
Formula
f(x; σ) = (x / σ²) exp(−x² / (2σ²)), x ≥ 0
F(x) = 1 − exp(−x² / (2σ²))
Mean = σ √(π/2), Variance = ((4 − π)/2) σ²
The Rayleigh distribution arises from the magnitude of a two-dimensional normal vector and is common in signal processing and wind speed modeling.
Full Description
The Rayleigh distribution describes the distribution of magnitudes when both orthogonal components follow independent, zero-mean, equal-variance normal distributions. It is frequently used in radar, communications, acoustics, and structural engineering to model signal envelopes, noise strength, or wave heights. The single parameter σ controls the spread: larger σ shifts the mode to higher values and increases variance.
Because the distribution is skewed right, mean exceeds mode. Engineers often rely on the Rayleigh model when directional components are isotropic and uncorrelated, such as turbulence or vibration amplitudes.
Frequently Asked Questions
How is σ estimated from data?
Use maximum likelihood: σ̂ = √(Σ xi² / (2n)).
Is Rayleigh a special case of another distribution?
Yes. It is a Chi distribution with two degrees of freedom and a Weibull distribution with shape parameter 2.
Can x be negative?
No. The Rayleigh distribution only applies for x ≥ 0.