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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 $\{1,2, \ldots, n\}$. The initial probability distribution of the particle is $\pi$ i.e., the probability that particle is in location $i$ is given by $\pi(i)$. In the first step, if the particle is at position $i$, it moves to one of the positions in $\{1,2, \ldots, i\}$ with uniform distribution; in the second step, if the particle is in location $j$, then it moves to one of the locations in $\{j, j+1, \ldots, n\}$ with uniform distribution. Suppose after two steps, the final distribution of the particle is uniform. What is the initial distribution $\pi?$

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

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(1)  $\pi$ is the initial distribution of particle being at location $i$.

$\implies \pi(1) + \pi(2) + \pi(3) + … + \pi(n) = 1$

(2) (First Step) When particle is in location $i$, it can move to any position in {$1,2,…,i$} with uniform probability ie probability of particle going to location $k$ is $\frac{1}{i}$ where $k \le i$.

(3) (Second Step) When particle is in location $k$ after first step, it can move to any position in {$k,k+1,…,n$} with uniform probability ie probability of particle going to location $j$ is $\frac{1}{n-k+1}$ where $j \ge k$.

As given, after completing both steps, the final distribution of particle is uniform in {$1,2,…,n$} with probability that particle is in location $j$ is $\pi’(j) = \frac{1}{n}$.

$\therefore \pi’(1) = \frac{1}{n}$

Now, the probability in terms of $\pi(i)$ of particle being at location $1$ is given by –

$\pi’(1) = \pi(1) * \frac{1}{1} * \frac{1}{n} + \pi(2) * \frac{1}{2} * \frac{1}{n} + \pi(3) * \frac{1}{3} * \frac{1}{n} + … + \pi(n) * \frac{1}{n} * \frac{1}{n}$

$\implies \frac{1}{n} = \pi(1) * \frac{1}{1} * \frac{1}{n} + \pi(2) * \frac{1}{2} * \frac{1}{n} + \pi(3) * \frac{1}{3} * \frac{1}{n} + … + \pi(n) * \frac{1}{n} * \frac{1}{n}$

$\implies 1 = \pi(1) * \frac{1}{1} + \pi(2) * \frac{1}{2} + \pi(3) * \frac{1}{3} + … + \pi(n) * \frac{1}{n}$

$\therefore \pi(1) + \pi(2) + \pi(3) + … + \pi(n) = \pi(1) * \frac{1}{1} + \pi(2) * \frac{1}{2} + \pi(3) * \frac{1}{3} + … + \pi(n) * \frac{1}{n}$

$\implies \pi(2) + \pi(3) + … + \pi(n) = \frac{\pi(2) }{2} + \frac{\pi(3)}{3} + … + \frac{\pi(n)}{n}$

This is only possible when $\pi(2) = \pi(3) = … = \pi(n) = 0$

$\implies \pi(1) = 1 \text{ and } \pi(i) = 0, \forall i \neq 1$