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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3541Start Needed for a 70% Upper-Hit Probability on [0, 6]For Brownian motion on [0, 6], at what starting point x is the probability of hitting 6 before 0 equal to 0.7?随机过程中等derivation未尝试面试订阅3542Upper-Hit Probability From x = 1 on [-3, 5]For Brownian motion started at x = 1 inside [-3, 5], what is the probability of hitting 5 before -3?随机过程中等derivation未尝试面试订阅3543Symmetric Interval Size for a 2/3 Upper-Hit ProbabilityBrownian motion starts at x = 1 inside [-L, L]. What L makes the probability of hitting +L before -L equal to 2/3?随机过程中等derivation未尝试面试订阅3544Lower Barrier Needed for a 40% Upper-Hit ProbabilityBrownian motion starts at x = 1 and has upper barrier 4 and lower barrier a < 1. What a makes the probability of hitting 4 before a equal to 0.4?随机过程中等derivation未尝试面试订阅3545Upper Barrier Needed for a 1/3 Hit Probability From x = 2Brownian motion starts at x = 2 with lower barrier 0 and upper barrier b > 2. What b makes the probability of hitting b before 0 equal to 1/3?随机过程中等derivation未尝试面试订阅3546Expected Exit Time From [0, 5] Started at 2For Brownian motion started at x = 2 inside [0, 5], what is the expected exit time from the interval?随机过程困难derivation未尝试面试订阅3547Expected Exit Time From [-2, 4] Started at 1For Brownian motion started at x = 1 inside [-2, 4], what is the expected exit time from the interval?随机过程困难derivation未尝试面试订阅