We can modify the Box-Muller method in Example 3.

1 to draw \(X\) and \(Y\) uniformly on the unit disc, \(\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2} \leqslant 1\right\}\), in the following way: Independently draw a radius \(R\) and an angle \(\Theta \sim \mathscr{U}(0,2 \pi)\), and return \(X=R \cos (\Theta), Y=R \sin (\Theta)\). The question is how to draw \(R\).

(a) Show that the cdf of \(R\) is given by \(F_{R}(r)=r^{2}\) for \(0 \leqslant r \leqslant 1\) (with \(F_{R}(r)=0\) and \(F_{R}(r)=\) 1 for \(r<0\) and \(r>1\), respectively).

(b) Explain how to simulate \(R\) using the inverse-transform method.

(c) Simulate 100 independent draws of \([X, Y]^{\top}\) according to the method described above.