# Name of PDF? - projecting uniform probability distribution on the unit circle to the x-axis

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.

• It appears to be exactly Arcsine(-r,r) ... en.wikipedia.org/wiki/… ... try $a=-r$, $b=r$ and compare. Commented Feb 1 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
Commented Feb 1 at 20:11
• This is a special case of the family of symmetric beta distributions described at stats.stackexchange.com/questions/85916.
– whuber
Commented Feb 1 at 20:20

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")

• 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
Commented Feb 1 at 19:12
• @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. Commented Feb 1 at 19:48
• Thanks, I appreciate the answer!
– SSD
Commented Feb 1 at 20:17