# Stochastics: Challenge Problem

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by Scott Alister McKinley on February 23, 2014

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 , does there exist a measure that satisfies the Detailed Balance condition.

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*Stochastic Processes*, Math 6030/7030

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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?

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.