The distribution of the distance between a point and its nearest neighbour in a spatial Poisson process. Also the distribution of the distance from the origin in n-dimensional space to the point (X₁, X₂,…, X_n), where X₁, X₂,…, X_n are independent normal variables (see normal distribution), each with expected value 0 and variance σ². The probability density function f is given by where σ>0 and Γ is the gamma function. The distribution has mean μ and variance V given by The distribution has mode . In the case n=2, the expressions for the mean and variance simplify to and σ²(4−π) respectively.

Rayleigh distribution. The shape of the distribution depends on the value of the parameters σ and n. The figure illustrates the dependence on n, with σ=1.