Random Walk Expected Time Interview Questions
Expected-time random walk interview prep for recurrence setup, plus-one terms, absorbing boundaries, fair walks, biased walks, and checks.
Candidates practicing expected hitting time recurrences in quant interviews.
Expected time needs a plus one
Let E_i be expected time to absorption from state i. After one move, one step has elapsed, so fair-walk equations usually look like E_i = 1 + 0.5E_{i-1} + 0.5E_{i+1}.
Boundary times are zero
If the process is already at an absorbing boundary, expected remaining time is zero. Write those boundary conditions before solving interior states.
Concrete example
For barriers at 0 and 2 starting at 1, E_0 = 0, E_2 = 0, and E_1 = 1 + 0.5E_0 + 0.5E_2 = 1. The walk absorbs after one step.
Compare to hitting probability
Expected time and hitting probability can use the same states but different equations. Probability recurrences omit the +1 because they are not counting elapsed steps.
Biased walk caution
For biased walks, transition probabilities change the weighted average. Do not solve a biased walk with symmetric coefficients unless the prompt says fair.
Common mistakes
Candidates often forget absorbing boundaries or the +1 term. If your expected time is negative or zero from a non-boundary state, the recurrence is broken.
Practice the pattern
Use the LeetQuidity curriculum and calibration to turn this topic into a focused practice plan.