🌊 Rayleigh Distribution Calculator
Enter the scale parameter (sigma) and a value of x to compute the Rayleigh distribution’s probability density, cumulative probability, mean, mode, and standard deviation.
What this calculator does
The Rayleigh distribution is a continuous probability distribution often used to model magnitudes, such as wind speed, wave heights, or the magnitude of a two-dimensional vector whose components are independent normal random variables. It is defined by a single scale parameter, sigma.
How the calculation works
The probability density function is f(x) = (x / sigma^2) x e^(-x^2 / (2 sigma^2)) for x >= 0. The cumulative distribution function, the probability that a Rayleigh random variable is less than or equal to x, is F(x) = 1 – e^(-x^2 / (2 sigma^2)). The distribution’s mean is sigma x sqrt(pi/2), its mode (most likely value) equals sigma itself, and its variance is (4 – pi)/2 x sigma^2.
Negative values of x are not valid for this distribution, so the calculator treats any negative input as 0. This tool is intended for general statistics and engineering education.