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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5981Total Stake Until the First SixYou roll a fair die repeatedly until the first time a 6 appears; let N be the number of rolls (including the winning roll). On each roll you independently collect a payoff X i with E[X i]=1.5, where the X i are i.i.d. and independent of the roll values. Compute E\! [\sum i=1 N X i ].概率简单derivation未尝试免费5982Variance of Fills Over a Binomial Number of QuotesOut of n=10 resting quotes, each fills independently with probability 0.3, so the number of fills N is Binomial(10,0.3). Each fill produces an i.i.d. PnL X i with E[X i]=2 and Var (X i)=9, independent of which quotes fill. For the stopped sum S N=\sum i=1 N X i, compute Var (S N).概率中等derivation未尝试免费5983Number of Trades to Cross a Profit TargetA strategy books i.i.d. positive profits X 1,X 2,\dots with E[X i]=2.5. Let N be the first time the running total S n=\sum i=1 n X i strictly exceeds 10; that is, N=\min\ n:S n>10\ . Assuming E[N]< , the expected overshoot is known to satisfy E[S N]=14. Use a Wald-style identity to compute E[N].概率中等derivation未尝试免费