Suppose we know how to generate bivariate observations $(X,Y)$ uniformly distributed on the unit disk centered at $(0,0)$. We are allowed to generate as many observations as desired. We want to find an unbiased estimator of $\pi$.
What I did is, generate $(X_1,Y_1),...,(X_n,Y_n)$ uniformly from the unit disk and consider $\theta_i=\tan^{-1}(\dfrac{Y_i}{X_i})$ which we know is uniformly distributed on $(0,2\pi)$.
So $E(\theta_i)=\pi$ for each $i$ which implies $\dfrac{1}{n}\sum_{i=1}^n \theta_i$ is an unbiased estimator of $\pi$.
Is this fine?
