A two-state chain in continuous time
For the two-state chain with \alpha=2 and \beta=3, find the stationary distribution and \mathbb{P}\bigl(X(t)=1\mid X(0)=0\bigr).
Worked answer (click to show)
The stationary distribution on states (0,1) is
\pi=\Bigl(\frac{\beta}{\alpha+\beta},\frac{\alpha}{\alpha+\beta}\Bigr)=\Bigl(\frac{3}{5},\frac{2}{5}\Bigr).
Starting from 0,
\mathbb{P}\bigl(X(t)=1\mid X(0)=0\bigr)=\frac{\alpha}{\alpha+\beta}\bigl(1-e^{-(\alpha+\beta)t}\bigr)=\frac{2}{5}\bigl(1-e^{-5t}\bigr).