The Rayleigh distribution is a continuous distribution often used to model the magnitude of a vector whose components are independent and normally distributed with mean zero. It is defined by a single scale parameter (σ) and is skewed right, starting at zero and rising to a peak before gradually tapering off.
Intuitively, the Rayleigh distribution describes the magnitude of a two-dimensional vector whose x and y components are independent and normally distributed around zero. Because magnitude is calculated as the distance from the origin, the resulting value is always non-negative, even though the components themselves can be positive or negative. This makes the Rayleigh distribution useful for modeling the overall strength of random effects like wind speed or signal strength, where only the total magnitude matters.
It is commonly used in engineering and signal processing, particularly for modeling noise or radar signal strength when only the magnitude is observed.
For example, the Rayleigh distribution can describe the distribution of wind speeds in meteorology or signal strength variations in wireless communication systems.
The graph below illustrates the Rayleigh distribution with a scale parameter of 2.0 and shows how it models wind speeds. The curve starts at zero, rises to a peak, and then gradually tapers off, reflecting how moderate wind speeds are most common, with both very low and very high speeds occurring less frequently.
