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651Fair Corridor Hit Probability 1A symmetric random walk starts at 2 on the integer line and stops when it first hits 0 or 7. What is the probability that it hits 7 before 0?概率简单derivation未尝试免费656Fair Corridor Exit Time 1A symmetric random walk starts at 1 and stops when it first hits 0 or 6. What is the expected stopping time?概率中等数值题未尝试免费661Lazy Walk Exit Time 1A lazy symmetric walk starts at 2 and, each period, moves +1 with probability 3/8, moves -1 with probability 3/8, and stays put with probability 1/4. It stops when it first hits 0 or 8. What is the expected stopping time?概率简单数值题未尝试免费667Biased Corridor Hit Probability 1A random walk starts at 2, moves +1 with probability 3/5 and -1 with probability 2/5, and stops when it first hits 0 or 7. What is the probability that it reaches 7 before 0?概率中等数值题未尝试免费673Scaled-Step Exit Time 3A fair random walk starts at 3 and moves by +3 or -3 with equal probability each step. It stops when it first hits -9 or 9. What is the expected stopping time?概率中等数值题未尝试免费674Scaled-Step Exit Time 2A fair random walk starts at 4 and moves by +4 or -4 with equal probability each step. It stops when it first hits 0 or 16. What is the expected stopping time?概率中等derivation未尝试免费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