wigner semi-circle distribution random numbers generation

Question

I am trying to generate random numbers in Wigner semi-circle distribution. Since this one does not have the analytical solution for the inverse function of the pdf. I wonder if anyone familiar with a standard way to generation random numbers (RNs) follow this distribution and what are the pros and cons for each method.

My initial guess from what I have researched is, I can use the rejection method or sample method on the uniform distributive random numbers to get the Wigner semi-circle one.

Previously, I have generated normally distributive RNs from uniformly distributive RNs using the Box-Muller transformation. Even I am not a statistics major, it was pretty straight forward process. However, I am having hard time grasp my head around other distributions, specifically this Wigner semi-circle.

Any instructions or source recommendations will be highly appreciated. Thank you.

Answer 1

[Following whuber's comments:] Since $$f(x)=\frac{2}{\pi R^2}\sqrt{R^2-x^2}$$the sub-graph of $f$ $$\mathcal S_R=\{(x,y);0\le y\le f(x)\}$$ is the half-disk of radius $R$. Thus, by the fundamental lemma of simulation, simulating $X\sim f$ is equivalent to simulating $(X,Y)$ uniformly on $\mathcal S$, which corresponds in spherical coordinates to simulating $$(\rho,\theta)\sim \frac{2}{\pi R}\rho \Bbb I_{(0,R)}(\rho) \Bbb I_{(0,\pi)}(\theta)$$ which is obvious:

1. simulate $U_r,U_a\sim\mathcal U(0,1)$
2. compute $X=R\, \sqrt{U_r} \cos (\pi U_a)$ [and do not compute $Y=R\, \sqrt{U_r} \sin (\pi U_a)$!]
3. return $X$

[and shows proximity with the Box-Mueller algorithm, although for the latter $\rho$ is distributed as an Exponential $\mathcal E(1/2)$].