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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. It is decidedly not the Arcsine distribution, as I have seen it mistakenly referred to occasionally!

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1 Answers1

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Well, I couldn't manage to find the Wikipedia page in question. It may be gone or it may never have existed. However, it has become clear to me that the distribution in question was indeed a generalized arcsine distribution with support on $[-r,r]$.

Please see my response to this question for further explanation.

https://math.stackexchange.com/a/4856249/228795

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