Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 7H / Solution

SetConditioning on $X_n$ gives the affine recurrenceIts fixed point is the first component of the stationary distribution,Thus $p_n-{\pi }_1=(1-\alpha -\beta {)}^n(p_0-{\pi }_1)$, and $p_0=1$ gives

Starting instead from state two gives the same recurrence with initial value zero. Therefore