 3

To generate a chi-squared variable with one degree of freedom by the Marsaglia polar method, repeatedly draw independent $V,W$ uniformly on $(-1,1)$ until $0<S=V^2+W^2<1$. ReturnThe accepted point is uniform on the unit disk; its squared radius is uniform on $(0,1)$ and independent of angle. The radial rescaling gives a standard normal coordinate $Z$, hence $Y\sim {\chi }_1^2$. The same accepted pair also supplies another independent standard normal coordinate, if needed. The proof stages below follow the four printed sampler labels.