For $X_n \in [0,1]$ and $r\in [0,1]$, we define the following martingale: $$ X_{n+1} = \begin{cases} 1 -r + rX_n & \text{with probability $X_n$} \\ rX_n & \text{with probability $1 - X_n$} \end{cases} $$
By the martingale convergence theorem, I know this converges almost surely. But how I expicitly compute the distribution of $X_{\infty}$? Is there a result I should keep in mind?