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## Homework Statement

Consider a random walk on a circle of N points, labeled {0,1,...,N-1}. Let the initial state be X = 0 and define T to be the first time all points have been visited at least once. Show that the distribution of X[T] (i.e. last unique position visited) is uniform over {1,...,N-1}.

## The Attempt at a Solution

I understand the basic formulas for Markov chains; including absorption probabilities, invariant distribution, and return times. The book makes an analogy to the Gambler's Ruin problem to show E[T] = N(N-1)/2. Still, I don't know where to start.

Thanks for the help,

Sam