# Distribution of Euclidean Distances Between Randomly Distributed Gaussian Points in n-Space

I am interested in the distribution of Eucildean distance between two points in a $$k$$-dimensional space.

Fortunately I came across the following paper which is pretty good in deriving the CDF and PDF:

Distribution of Euclidean Distances Between Randomly Distributed Gaussian Points in n-Space

Unfortunately, I have troubles understanding some of the notations.

In particular the authors define the two points in a k-dimensional space as $$\Gamma = (\gamma_1, \ldots, \gamma_k)$$ and $$\Psi = (\psi_1, \ldots, \psi_k)$$. However, later on the CDF is derived the following: $$F(R, k) = 1 - \frac{\Gamma(\frac{k}{2}, \frac{R^2}{4})}{\Gamma(\frac{k}{2})}$$ in which $$R$$ seems to be defined as "the probability that the CDF is less than or equal to R". Taking aside the definition of $$R$$, the set $$\Gamma$$ seems suddenly to be a function with one or two inputs. The paper does not contain a definition of this function.

Hence my question, what does $$\Gamma(x, y)$$ and $$\Gamma(x)$$ return in this context?

Note: I tried to contact the original authors with out much success.

• Perhaps the (generalized) Gamma functions - en.wikipedia.org/wiki/Incomplete_gamma_function? Commented Nov 6, 2019 at 3:51
• what if the Gaussians were not standard normal but some N(0,sigma^2). mean still 0 but not a unit variance. what would the result be in terms of the distrib? hope it doesnt change a tonne! Commented Oct 19, 2021 at 17:48

As Calvin Lin suggested in the comments,

• $$\Gamma(x) = \int\limits_0^\infty t^{x-1} e^{-t}\,dt$$, the Gamma function, which for integer $$x$$ is $$\Gamma(x)=(x-1)!$$
• $$\Gamma(x,y) = \int\limits_y^{\infty} t^{x-1}e^{-t}\,dt$$, the upper incomplete Gamma function

The work in your link looks a little excessive. It is looking at the distribution of the distance between two independent standard multivariate normal random vectors. If this is $$k$$-dimensional and the distance is $$D$$, then you could say that

• $$\frac12 D^2$$ has a chi-squared distribution with $$k$$ degrees of freedom
• $$D^2$$ has a gamma distribution with shape parameter $$\frac k2$$ and scale parameter $$4$$
• $$\frac1{\sqrt 2} D$$ has a chi distribution with $$k$$ degrees of freedom
• $$D$$ has a generalized gamma distribution with parameters $$p = 2$$, $$d = k$$, and scale parameter $$2$$

and the moments of these are well known

$$F(R,k)=\mathbb{P}(\Vert X-Y\Vert\le R),$$ where $$X,\,Y\in\mathbb{R}^k$$ and $$X,Y\sim N(0,I_k).$$ That is, it is the cdf of the distance between two $$k$$-dimensional standard gaussian random variables. Secondly, as pointed out in the comments, $$\Gamma\left(\dfrac{k}{2},\dfrac{R^2}{4}\right)$$ denotes the incomplete gamma function, and $$\Gamma\left(\dfrac{k}{2}\right)$$ is the standard gamma function.