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Fix $n\geq 4.$ Suppose there is a particle that moves randomly on the number line, but never leaves the set $\{1,2,\dots,n\}.$ Let the initial probability distribution of the particle be denoted by $\overrightarrow{\pi}.$ In the first step, if the particle is at position $i,$ it moves to one of the positions in $\{1,2,\dots,i\}$ with uniform distribution; in the second step, if the particle is in location $j,$ then it moves to one of the locations in $\{j,j+1,\dots,n\}$ with uniform distribution.Suppose after two steps, the final ditribution of the particle is uniform. What is the initial distribution $\overrightarrow{\pi}?$

  1. $\overrightarrow{\pi}$ is not unique
  2. $\overrightarrow{\pi}$ is uniform
  3. $\overrightarrow{\pi(i)}$ is non-zero for all even $i$ and zero otherwise
  4. $\overrightarrow{\pi}(1) = 1$ and $\overrightarrow{\pi}(i) = 0$ for $i\neq 1$
  5. $\overrightarrow{\pi}(n) = 1$ and $\overrightarrow{\pi}(i) = 0$ for $i\neq n$
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