Paper 3, Section I, H
Part IB, 2013
Prove that if a distribution is in detailed balance with a transition matrix then it is an invariant distribution for .
Consider the following model with 2 urns. At each time, one of the following happens:
with probability a ball is chosen at random and moved to the other urn (but nothing happens if both urns are empty);
with probability a ball is chosen at random and removed (but nothing happens if both urns are empty);
with probability a new ball is added to a randomly chosen urn,
where and . State denotes that urns 1,2 contain and balls respectively. Prove that there is an invariant measure
Find the proportion of time for which there are balls in the system.