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3690Why OST and Reflection Solve Different Brownian QuestionsWhy should OST problems not be confused with reflection-principle problems even though both often involve Brownian barriers?随机过程中等essay未尝试面试订阅5901Expected Rounds to Double a Kelly BankrollA gambler bets the Kelly fraction on an even-money coin with win probability p=0.6 every round, so log-wealth is a random walk with positive drift. Let G be the per-round expected log-growth (the maximal Kelly growth rate). Using an optional-stopping argument on a suitable martingale, estimate the expected number of rounds until wealth first doubles. You may ignore overshoot past the doubling level.概率困难数值题未尝试面试订阅5962Time to Climb One Step (Biased Walk)A walk starts at 0 and each step moves +1 with probability 2/3 and -1 with probability 1/3. Let T be the first time it reaches +1. Find E[T].概率中等数值题未尝试免费5963Wald's Identity from Optional StoppingLet X 1,X 2,... be i.i.d. with mean 4, and let N be a stopping time (with respect to the X's) with E[N]=10. Using the martingale M n = sum i<=n X i - 4n and optional stopping, find E[X 1+...+X N].概率简单数值题未尝试免费5964Polya Urn Limiting FractionAn urn starts with 1 red and 2 blue balls. Each step a ball is drawn uniformly at random, observed, and returned together with one additional ball of the same color. Let R n/T n be the fraction of red balls after n draws. This fraction is a bounded martingale converging to a limit L. Using optional stopping / martingale convergence, find E[L].概率中等数值题未尝试免费5965Branching Process Extinction ProbabilityA Galton-Watson branching process starts with one individual. Each individual independently has 0 offspring with probability 1/4, 1 offspring with probability 1/4, and 2 offspring with probability 1/2. Let q be the extinction probability. Using that q Z n is a martingale (where Z n is the generation-n population), find q.概率困难数值题未尝试面试订阅5966Symmetric Exit ValueA symmetric simple random walk starts at 0 and stops the first time it reaches +3 or -3. By symmetry and optional stopping, what is the expected value of the walk at the stopping time, E[S T]?概率简单数值题未尝试免费5967Doubling Strategy and Optional Stopping FailureA gambler starts with 0 net and makes fair 1-doubling bets (bet 1, then 2, then 4, ...) on a sequence of fair coin flips, stopping the first time they win a single flip (guaranteeing +1 net). Let T be that stopping time. Compute E[net wealth at T], and explain whether E[net at T] equals net at time 0 as naive optional stopping would suggest.概率中等数值题未尝试免费5968Waiting Time for the Pattern HTHHA fair coin is flipped repeatedly. Using a martingale (gambling-team) argument, find the expected number of flips until the pattern H, T, H, H first appears.概率中等数值题未尝试免费5969Three Consecutive SixesA fair six-sided die is rolled repeatedly. Using a martingale (gambling-team) argument, find the expected number of rolls until three sixes appear in a row.概率中等数值题未尝试免费5970Ballot Problem via MartingaleIn an election candidate A receives 7 votes and candidate B receives 3 votes; the 10 votes are counted in a uniformly random order. Using a martingale / optional stopping argument, find the probability that A is strictly ahead of B throughout the entire count.概率困难数值题未尝试面试订阅5971Expected Duration of Biased Gambler's RuinA walk starts at 2, moves +1 with probability 2/3 and -1 with probability 1/3, and stops on hitting 0 or 5. First find the probability it exits at 5, then use the linear-drift martingale to find the expected duration E[T].概率困难数值题未尝试面试订阅5972Hitting the Opposite Node on a CycleA token does a symmetric random walk on the 6 vertices of a cycle (labelled 0..5, each step moving to one of the two adjacent vertices with probability 1/2 each). Starting at vertex 0, find the expected number of steps to first reach vertex 3, the diametrically opposite vertex.概率简单数值题未尝试免费5973Ruin Duration With Holding TiesA score starts at 3 and each round goes +1 with probability 0.3, -1 with probability 0.3, and stays the same (a tie) with probability 0.4. The game ends when the score first reaches 0 or 8. Find the expected number of rounds until the game ends.概率中等数值题未尝试免费5974Azuma Bound on a Bounded MartingaleLet M 0=0, M 1, M 2, ... be a martingale whose increments satisfy |M k - M k-1 | <= 1 for all k. Using the Azuma-Hoeffding inequality, give the best upper bound it provides on P(M 100 >= 20).概率困难数值题未尝试面试订阅5975Expected Sample Size of a Sequential Boundary TestIndependent draws X 1, X 2, ... each equal +1 with probability 0.6 and -1 with probability 0.4, accumulated into S n. Sampling stops when S n first reaches +5 or -5. The exit probabilities are P(exit +5)=0.883636 and P(exit -5)=0.116364. Using the linear-drift (Wald-Wolfowitz) martingale, find the expected number of draws E[N].概率中等数值题未尝试免费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