Editing N-sphere (section)

== Probability distributions ==

=== Uniformly at random on the {{math|(''n'' − 1)}}-sphere ===
[[File:2sphere-uniform.png|thumbnail|A set of points drawn from a uniform distribution on the surface of a unit {{math|2}}-sphere, generated using Marsaglia's algorithm.]]

To generate uniformly distributed random points on the unit {{tmath|(n-1)}}-sphere (that is, the surface of the unit {{tmath|n}}-ball), {{harvtxt|Marsaglia|1972}} gives the following algorithm.

Generate an {{tmath|n}}-dimensional vector of [[normal distribution|normal deviates]] (it suffices to use {{tmath|N(0, 1)}}, although in fact the choice of the variance is arbitrary), {{tmath|\mathbf x {{=}} (x_1, x_2, \ldots, x_n)}}. Now calculate the "radius" of this point:

:<math>r=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}.</math>

The vector {{tmath|\tfrac 1r \mathbf x}} is uniformly distributed over the surface of the unit {{tmath|n}}-ball.

An alternative given by Marsaglia is to uniformly randomly select a point {{tmath|\mathbf x {{=}} (x_1, x_2, \ldots, x_n)}} in the unit [[hypercube|{{mvar|n}}-cube]] by sampling each {{tmath|x_i}} independently from the [[continuous uniform distribution|uniform distribution]] over {{tmath|(-1, 1)}}, computing {{tmath|r}} as above, and rejecting the point and resampling if {{tmath|r \geq 1}} (i.e., if the point is not in the {{tmath|n}}-ball), and when a point in the ball is obtained scaling it up to the spherical surface by the factor {{tmath|\tfrac 1r }}; then again {{tmath|\tfrac 1r \mathbf x}} is uniformly distributed over the surface of the unit {{tmath|n}}-ball. This method becomes very inefficient for higher dimensions, as a vanishingly small fraction of the [[unit cube]] is contained in the sphere. In ten dimensions, less than 2% of the cube is filled by the sphere, so that typically more than 50 attempts will be needed. In seventy dimensions, less than <math>10^{-24}</math> of the cube is filled, meaning typically a trillion quadrillion trials will be needed, far more than a computer could ever carry out.

=== Uniformly at random within the ''n''-ball ===
With a point selected uniformly at random from the surface of the unit {{tmath|(n-1)}}-sphere (e.g., by using Marsaglia's algorithm), one needs only a radius to obtain a point uniformly at random from within the unit {{tmath|n}}-ball.  If {{tmath|u}} is a number generated uniformly at random from the interval {{tmath|[0, 1]}} and {{tmath|\mathbf x}} is a point selected uniformly at random from the unit {{tmath|(n-1)}}-sphere, then {{tmath|u^{1/n} \mathbf x}} is uniformly distributed within the unit {{tmath|n}}-ball.

Alternatively, points may be sampled uniformly from within the unit {{tmath|n}}-ball by a reduction from the unit {{tmath|(n+1)}}-sphere. In particular, if {{tmath|(x_1, x_2, \ldots, x_{n+2})}} is a point selected uniformly from the unit {{tmath|(n+1)}}-sphere, then {{tmath|(x_1, x_2, \ldots, x_n)}} is uniformly distributed within the unit {{tmath|n}}-ball (i.e., by simply discarding two coordinates).<ref>{{cite report|first1=Aaron R. | last1=Voelker | first2=Jan | last2=Gosmann | first3=Terrence C. | last3=Stewart | title=Efficiently sampling vectors and coordinates from the n-sphere and n-ball | year=2017 | publisher=Centre for Theoretical Neuroscience | url=http://compneuro.uwaterloo.ca/publications/voelker2017.html | doi=10.13140/RG.2.2.15829.01767/1}}</ref>

If {{tmath|n}} is sufficiently large, most of the volume of the {{tmath|n}}-ball will be contained in the region very close to its surface, so a point selected from that volume will also probably be close to the surface.  This is one of the phenomena leading to the so-called [[curse of dimensionality]] that arises in some numerical and other applications.

=== Distribution of the first coordinate ===
Let {{tmath|y {{=}} x_1^2 }} be the square of the first coordinate of a point sampled uniformly at random from the {{tmath|(n-1)}}-sphere, then its probability density function, for <math>y\in [0, 1]</math>, is 

<math display="block">
\rho(y) = \frac{\Gamma\bigl(\frac{n}{2} \bigr)}{\sqrt\pi \; \Gamma\bigl(\frac{n-1}{2}\bigr)} (1-y)^{(n-3)/2}y^{-1/2}.
</math>

Let <math>z = y/N</math> be the appropriately scaled version, then at the <math>N\to \infty</math> limit, the probability density function of <math>z</math> converges to <math> (2\pi ze^z)^{-1/2}</math>. This is sometimes called the Porter–Thomas distribution.<ref>{{Citation |last=Livan |first=Giacomo |title=One Pager on Eigenvectors |date=2018 |url=https://doi.org/10.1007/978-3-319-70885-0_9 |work=Introduction to Random Matrices: Theory and Practice |pages=65–66 |editor-last=Livan |editor-first=Giacomo |access-date=2023-05-19 |series=SpringerBriefs in Mathematical Physics |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-70885-0_9 |isbn=978-3-319-70885-0 |last2=Novaes |first2=Marcel |last3=Vivo |first3=Pierpaolo |editor2-last=Novaes |editor2-first=Marcel |editor3-last=Vivo |editor3-first=Pierpaolo}}</ref>