Variance of the time for a drifting walk to reach 7
Question
A random walk on the integers starts at ; each step goes up by with probability and down by with probability . Let be the number of steps until the walk first reaches . What is ? Answer if it is infinite.
Short answer
. is a sum of independent, identically distributed passage times, each from one level to the next. A single passage has variance when , so .
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Open the worked solutionLessons behind this question
- Gambler's Ruin and First Passage Times
- Conditional Probability, Total Probability, and Bayes' Rule
- Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating
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