# Stochastics: Challenge Problem

If you’re struggling with understanding the notion of detailed balance, try out this quick challenge problem.

• Consider RW on the graph pictured below.  For what value(s) of $p$, does there exist a measure $\mu$ that satisfies the Detailed Balance condition.

## 2 thoughts on “Stochastics: Challenge Problem”

1. Franco says:

Can anyone explain how to do this problem? I know detailed balance and how to determine whether a MC meets the DB condition, but given that we do not know p in this problem, how would you calculate all the possible p’s for which DB is met?

2. Josh G says:

My thought is to assume (WLOG) that p > .5. Then in order to satisfy detailed balance you need each subsequent pi(x) larger than the last, which results in a contradiction since it is a loop and 6 returns to 1.