Two-State Random Walk Distribution

In this one-dimensional walk, the walker either moves forward or backward (up or down, or through any two-state abstraction), but never stays still. For any possible walk of steps there are possible paths. Each of these paths corresponds to a probability distribution on the states where the probability assigned to a state is the fraction of time that the path spends in that state. The left plot shows the path and the right plot shows the corresponding probability distribution.

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