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,…, Xn), where X1, X2,…, Xn are independent normal variables, each with expectation 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 σ√n-1. In the case n=2, the expressions for the mean and variance simplify to σ√½π and ½σ2(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.
Subjects: Probability and Statistics.