"Dason: Like how would you propose something simpler... like turning the equation for a circle into a pdf: x^2 + y^2 = 1"

And because I'm in a procrastinating mood (I've got like a bazillion things to do before the end of the month) I thought about it during the evening and I think I came up with an answer. So I submit it for his verification

This is what I came up with:

Based on what I remember back in the day:

https://blogs.ubc.ca/math105/continuous-random-variables/the-pdf/

The challenge is really to turn \(x^{2}+y^{2}=1\) into having 2 (important) properties: It has to be non-negative, and it should integrate to 1.

So, for \(x^{2}+y^{2}=1\) to be positive I only need to remove the bottom half (the negative part) of the circle in the (X,Y) plane. Which I can do by only taking the positive square root part and restricting the domain as:

\(y=\sqrt{1-x^{2}}, |x|\leq1\)

So that takes care of the first part. The second part is scaling the function so that it integrates to 1. O.K, well, I know that the area of this particular semi-circle is \(\pi/2\). So I can just re-scale it by the reciprocal \(2/\pi\).

And I guess the final step is to put it all together in a piece-wise function like:

\(

y= \begin{cases}

2/\pi\sqrt{1-x^{2}} & |x|\leq1\\

0 & otherwise \\

\end{cases}

\)

So... yeah. I guess that's what I'd do to turn the unit circle into a valid PDF. ¯\_(ツ)_/¯[/MATH][/MATH]