In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distribution is named after Lord Rayleigh (/ˈreɪli/).

A Rayleigh distribution is observed when the overall magnitude of a vector in the plane is related to its directional components. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed in two dimensions. Assuming that each component is uncorrelated, normally distributed with equal variance, and zero mean, then the overall wind speed (vector magnitude) will be characterized by a Rayleigh distribution. (In reality these assumptions are rarely even approximately satisfied, however, so the Weibull distribution is more frequently used in practice.) A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are independently and identically distributed Gaussian with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.

The probability density function of the Rayleigh distribution is

where ${\displaystyle \sigma }$ is the scale parameter of the distribution. The cumulative distribution function is

for ${\displaystyle x\in [0,\infty ).}$

Consider the two-dimensional vector ${\displaystyle Y=(U,V)}$ which has components that are bivariate normally distributed, centered at zero, with equal variances ${\displaystyle \sigma ^{2}}$, and independent. Then ${\displaystyle U}$ and ${\displaystyle V}$ have density functions

Let ${\displaystyle X}$ be the length of ${\displaystyle Y}$. That is, ${\displaystyle X={\sqrt {U^{2}+V^{2}}}.}$ Then ${\displaystyle X}$ has cumulative distribution function

where ${\displaystyle D_{x}}$ is the disk