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# 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 Comments Post a comment
1. Franco

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?

February 25, 2014
2. Josh G

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.

February 26, 2014