Prove that if a distribution $\pi$ is in detailed balance with a transition matrix $P$ then it is an invariant distribution for $P$.

Consider the following model with 2 urns. At each time, $t=0,1, \ldots$ one of the following happens:

• with probability $\beta$ a ball is chosen at random and moved to the other urn (but nothing happens if both urns are empty);

• with probability $\gamma$ a ball is chosen at random and removed (but nothing happens if both urns are empty);

• with probability $\alpha$ a new ball is added to a randomly chosen urn,

where $\alpha+\beta+\gamma=1$ and $\alpha<\gamma$. State $(i, j)$ denotes that urns 1,2 contain $i$ and $j$ balls respectively. Prove that there is an invariant measure

$$\lambda_{i, j}=\frac{(i+j) !}{i ! j !}(\alpha / 2 \gamma)^{i+j}$$

Find the proportion of time for which there are $n$ balls in the system.