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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3191Total PnL Until a Geometric Number of FillsLet X 1,X 2,\dots be i.i.d. increments with E[X i]=3 and Var (X i)=5. Let N be independent of the increments and distributed as Geometric( 1 4 ) on 1,2,\dots . For the stopped sum S N=\sum i=1 N X i, compute E[S N] and Var (S N).概率中等derivation未尝试面试订阅3192Aggregate Slippage Over a Poisson Number of OrdersLet X 1,X 2,\dots be i.i.d. increments with E[X i]=2 and Var (X i)=3. Let N be independent of the increments and distributed as Poisson(4). For the stopped sum S N=\sum i=1 N X i, compute E[S N] and Var (S N).概率中等derivation未尝试面试订阅3193Total Cost Over a Negative-Binomial HorizonLet X 1,X 2,\dots be i.i.d. increments with E[X i]=4 and Var (X i)=6. Let N be independent of the increments and distributed as NegativeBinomial(r=3, p= 2 5 ). For the stopped sum S N=\sum i=1 N X i, compute E[S N] and Var (S N).概率中等derivation未尝试面试订阅3201Expected Trials to Reach 5 SuccessesIndependent Bernoulli trials succeed with probability 2 5 . Let T be the first time the cumulative number of successes reaches 5. Use Wald-style reasoning to compute E[T].概率中等derivation未尝试面试订阅3206Variance of Trials to Reach 5 SuccessesIndependent Bernoulli trials succeed with probability 2 5 . Let T be the first time the cumulative number of successes reaches 5. Use Wald-style second-moment reasoning to compute Var (T).概率困难derivation未尝试面试订阅3212Second Moment of Centered Sum at a Poisson HorizonLet X 1,X 2,\dots be i.i.d. with mean and variance 3. Let N be independent of the increments and distributed as Poisson(4). Show that for the centered stopped sum M N=\sum i=1 N (X i- ), one has E[M N 2] equal to what value?概率中等derivation未尝试面试订阅3214Centered Slippage Variance Under Random StoppingLet X 1,X 2,\dots be i.i.d. with mean and variance 4. Let N be independent of the increments and distributed as Geometric( 1 3 ). Show that for the centered stopped sum M N=\sum i=1 N (X i- ), one has E[M N 2] equal to what value?概率中等derivation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