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Variance of the time for a drifting walk to reach 7

Question

A random walk on the integers starts at 00; each step goes up by 11 with probability p=23p=\tfrac23 and down by 11 with probability 13\tfrac13. Let TT be the number of steps until the walk first reaches 77. What is Var⁡(T)\operatorname{Var}(T)? Answer −1-1 if it is infinite.

Short answer

168168. TT is a sum of 77 independent, identically distributed passage times, each from one level to the next. A single passage has variance 4p(1−p)(2p−1)3=24\tfrac{4p(1-p)}{(2p-1)^3}=24 when p=23p=\tfrac23, so Var⁡(T)=7⋅24=168\operatorname{Var}(T)=7\cdot24=168.

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