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Consider a uniform probability distribution on a circle of radius r, i.e. $\{(x,y) \in \mathbb{R}^2: x^2 + y^2 = r^2 \}$.If we wish to project onto the x-axis, we can consider each point on the circle to contribute $\delta(x-rcos(\theta))$. Then averaging over all possible values of $\theta$ yields the following distribution:

$Pr(x) = \frac{1}{\pi\sqrt{r^2-x^2}}$

Intriguingly, the Fourier transform of Pr(x) happens to be the zero order Bessel function of the first kind, i.e. $J_0(2\pi r f)$

Question: What is this distribution Pr(x) called?

I distinctly recall it having a Wikipedia page dedicated to it, and it being named after some mathematician. EDIT: It is equivalent to the Arcsine distribution on the interval (-r,r), but I could swear I've seen it with its own page under a different name.

SSD
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    It appears to be exactly Arcsine(-r,r) ... https://en.wikipedia.org/wiki/Arcsine_distribution#Arbitrary_bounded_support ... try $a=-r$, $b=r$ and compare. – Glen_b Feb 01 '24 at 19:43
  • You're exactly correct, and I've edited the post to reflect that. I just could swear I've seen it under a separate wikipedia article where the C.F. is explicitly listed as a Bessel function. Maybe I'm on a wild goose chase. Cheers – SSD Feb 01 '24 at 20:11
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    This is a special case of the family of symmetric beta distributions described at https://stats.stackexchange.com/questions/85916. – whuber Feb 01 '24 at 20:20

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If $X$ and $Y$ are two independent standard Gaussian distributions, then the random point $\left(\frac{X}{\sqrt{X^2+Y^2}}, \frac{Y}{\sqrt{X^2+Y^2}}\right)$ has the uniform distribution on the unit circle. This is a particular case of a projected normal distribution, which is more generally defined for $(X,Y)$ following some bivariate Gaussian distribution. Of course you have to scale by $r$ to get the uniform distribution on the centered circle with radius $r$.

However, it is true that it is an Arcsine distribution, but the one with support $(-1, 1)$:

library(uniformly) # to generate uniform points on the circle
library(distr)     # to generate the arcsine distribution on (-1, 1)

uniform points on the unit circle


rcirc <- runif_on_sphere(1000, 2)


arcsine-distributed points on (-1, 1)


A <- Arcsine()
rasin <- r(A)(1000)


comparison of the empirical distribution functions


plot(ecdf(rcirc[, 1]))
lines(ecdf(rasin), col = "blue")
Stéphane Laurent
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  • To clarify - I'm not concerned with the distribution on the unit circle - I'm concerned with the distribution when project onto the X-axis. So, marginalizing out the y dependence. Then you end up with what in my question I refer as $Pr(x)$ – SSD Feb 01 '24 at 19:12
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    @SSD Yes, I understood. But there's nothing to marginalize: one can simply say that this distribution is the first coordinate of the projected normal. I also edited my answer to add something. – Stéphane Laurent Feb 01 '24 at 19:48
  • Thanks, I appreciate the answer! – SSD Feb 01 '24 at 20:17