Please imagine a discrete random walk on a one-dimensional lattice.  The lattice consists of a set of $L$ positions, $(x_0, x_1, ..., x_L) \in L$, where $x_0$ is the initial position of the walk (as well as a reflecting boundary), and $x_L$ is absorbing.  

For each position in the walk, we assign one of $N$ jump probabilities (forward, $p_k$, backward, $(1-p_k)$) from a set $P$, where $(p_1, p_2, ..., p_N) \in P$.
 
For the duration of the random walk, until the absorbing target $x_L$ is reached, what is the mean occupancy of the a given position in the one-dimensional lattice, $x_k$?  I'm hoping to find an efficient method to compute an exact solution.

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The 'reverse' question may also be interesting - 

Let $M$, $(m_1, m_2, ..., m_L) \in M$ be the set of mean occupancy values for each position in the one-dimensional lattice, $(x_0, x_1, ..., x_L) \in L$.  Is there an efficient method to uniquely reconstruct the values for the set of jump probabilities, $(p_1, p_2, ..., p_N) \in P$ (as defined above), for each position in the lattice, $x_k$?