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5976Expected Maximum Before RuinA fair simple random walk starts at 2 and is absorbed when it first hits 0 or 5. Let H be the highest level the walk reaches over its whole path (the running maximum at absorption). Find E[H].概率困难数值题未尝试面试订阅5977Optional Stopping on a Product MartingaleLet X 1, X 2, ... be i.i.d. fair +-1 steps and define the product P n = prod i=1 n (1 + (1/2) X i), with P 0 = 1. Let N be any almost-surely finite stopping time. Treating P n as a martingale, what is E[P N]?概率中等数值题未尝试免费5978First Passage Probability for a Lazy Biased WalkA lazy walk starts at 1. Each step it moves +1 with probability 0.3, -1 with probability 0.2, and stays put with probability 0.5. It stops on first reaching +4 or -4. Find the probability it exits at +4.概率中等数值题未尝试免费5979Three-Level Exit With a Non-Absorbing BarrierA fair simple random walk starts at 0. Three levels are marked: -3, +2, and +6. Only the two extreme levels, -3 and +6, are absorbing; the walk passes freely through +2. Find the probability the walk is absorbed at -3, and the expected absorption value E[S T].概率困难数值题未尝试面试订阅5980Random Walk With Rest PeriodsA process evolves in rounds. Each round, independently, with probability 1/2 the walker rests (position unchanged) and with probability 1/2 it takes a step that is +1 or -1 with equal odds. The walker stops at the first round in which it completes its 8th actual (non-rest) step. Let S be the position at that stopping time. Find E[S 2].概率中等数值题未尝试免费