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3538Why an Endpoint Cap Changes the Risk So MuchWhy can adding a condition such as W t <= b dramatically shrink the probability relative to the unconditional touch event?随机过程中等essay未尝试面试订阅3539Why Reflection Helps With Barriers but Not Every FunctionalWhy is reflection so powerful for barrier events yet much less useful for path functionals like time averages?随机过程中等essay未尝试面试订阅3540Reflection Versus Optional StoppingIn one paragraph, contrast what reflection principle and optional stopping are each best at in Brownian barrier problems.随机过程中等essay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3548Symmetric Half-Width Needed for Expected Exit Time 8Brownian motion starts at x = 1 inside [-L, L]. What L makes the expected exit time equal to 8?随机过程困难derivation未尝试面试订阅3549Lower Barrier Needed for Expected Exit Time 12Brownian motion starts at x = 2 with upper barrier 5 and lower barrier a. What a makes the expected exit time equal to 12?随机过程困难derivation未尝试面试订阅3550Upper Barrier Needed for Expected Exit Time 6 From x = 1.5Brownian motion starts at x = 1.5 with lower barrier 0 and upper barrier b. What b makes the expected exit time equal to 6?随机过程困难derivation未尝试面试订阅3551Average Upper-Hit Probability With Uniform Start on [0, 6]If the starting point X is uniform on [0, 6], what is the average probability of hitting 6 before 0?随机过程困难derivation未尝试面试订阅3552Average Exit Time With Uniform Start on [0, 6]If the starting point X is uniform on [0, 6], what is the expected exit time from [0, 6]?随机过程困难derivation未尝试面试订阅3554Average Exit Time With Uniform Start on [-3, 3]If X is uniform on [-3, 3], what is the expected exit time from [-3, 3]?随机过程困难derivation未尝试面试订阅3555Conditional Upper-Hit Probability From the Upper Half of [0, 8]If X is uniform on [0, 8] conditioned on X > 4, what is the average probability of hitting 8 before 0?随机过程困难derivation未尝试面试订阅3556Interval Width When the Start Is One-Third of the Way UpBrownian motion starts at x = b/3 inside [0, b]. What b makes the expected exit time equal to 8?随机过程中等derivation未尝试面试订阅3557Expected Exit Time When Upper-Hit Probability Is 2/3 on [0, 9]Brownian motion starts somewhere in [0, 9] such that the probability of hitting 9 before 0 is 2/3. What is the expected exit time?随机过程中等derivation未尝试面试订阅3558Symmetric Interval and Start Matching Two TargetsBrownian motion starts at x inside [-L, L]. If the probability of hitting +L before -L is 0.75 and the expected exit time is 12, what is x?随机过程中等derivation未尝试面试订阅