Given two vector valued RVs $ X,Y \in S_1$ (the unit circle) that are uniformly distributed. We choose a fixed vector $a \in S_1$, e.g. in x-direction: $a=e_x$.
We then sample once from both distributions and obtain $x_1, y_1 \in S_1$ in the first run. The resulting vectors $z$ are then chosen as follows: $$z_1=x_1 \text{ if } |a \cdot x_1| \ge |a \cdot y_1| \text{ and } z_1=y_1 \text{ if } |a \cdot x_1| < |a \cdot y_1|$$
What is the resulting density $\rho(z|a)$ of the vectors $z$? What integrals do I have to evaluate and why?
Any help is highly appreciated.