Expected Hitting Time of a Birth-Death Chain
A Markov chain has states
, 2, \ldots, n$. From state $i$, the chain moves to $i+1$ with probability $p$, to $i-1$ with probability $q$, and stays at $i$ with probability
- p - q$. State 1 has a reflecting boundary (a leftward step from state 1 keeps you at state 1).
Let $h_i$ denote the expected number of steps to reach state $n$ starting from state $i$.
1. Set up and solve the recurrence for $h_i$ in terms of $p$ and $q$ (with $p \ne q$).
2. Solve the symmetric case $p = q = 1/2$ and express $h_1$ as a closed-form function of $n$.
3. How does the "stay" probability
- p - q$ affect the answer?
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