Table of Contents
What Is the Rayleigh Distribution?
The Rayleigh distribution is a continuous probability distribution for non-negative random variables. It arises naturally when the magnitude of a two-dimensional vector is computed from two independent, identically distributed, zero-mean normal components. Named after Lord Rayleigh, it is widely used in physics, engineering, and communications.
Common applications include modeling wind speed magnitudes, the lifetime of certain types of components, scattered signal amplitudes in radar and communications (Rayleigh fading), and the distribution of distances from the origin in a random walk in two dimensions.
Formulas
Properties
| Property | Formula |
|---|---|
| Mean | σ × sqrt(π/2) ≈ 1.2533 × σ |
| Median | σ × sqrt(2 × ln(2)) ≈ 1.1774 × σ |
| Mode | σ |
| Variance | (4 - π) / 2 × σ² ≈ 0.4292 × σ² |
| Skewness | 2√π(π - 3) / (4 - π)3/2 ≈ 0.6311 |
| Support | x ∈ [0, +∞) |
Frequently Asked Questions
When should I use the Rayleigh distribution?
Use the Rayleigh distribution when modeling the magnitude (distance) of a 2D vector whose components are independent, zero-mean, equal-variance normal random variables. Practical examples include wind speed at a location, scattered signal envelope in multipath channels, and the distance of darts from the center of a target.
How is the Rayleigh distribution related to the normal distribution?
If X and Y are independent normal random variables with mean 0 and variance σ², then the magnitude R = sqrt(X² + Y²) follows a Rayleigh distribution with parameter σ. It is also a special case of the Weibull distribution (shape parameter = 2) and the chi distribution (k = 2).