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3516Touch Probability for Barrier 1 Over Horizon 1For standard Brownian motion, what is P(max 0<=s<=1 W s >= 1)?随机过程中等derivation未尝试面试订阅3518Barrier Giving Touch Probability 0.1 Over Horizon 9For standard Brownian motion over horizon t = 9, what barrier a gives touch probability 0.1?随机过程中等derivation未尝试面试订阅3519Barrier Giving Touch Probability 0.02 Over Horizon 0.25For standard Brownian motion over horizon t = 0.25, what barrier a gives touch probability 0.02?随机过程中等derivation未尝试面试订阅3520Horizon Needed for Touch Risk 0.2 Under Barrier 2For standard Brownian motion, what horizon t gives touch probability P(M t >= 2) = 0.2?随机过程中等derivation未尝试面试订阅3521Horizon Needed for Touch Risk 0.05 Under Barrier 1.8For standard Brownian motion, what horizon t gives touch probability P(M t >= 1.8) = 0.05?随机过程中等derivation未尝试面试订阅3522Barrier Matching a Unit-Horizon Risk Across Horizon 4A barrier of 1.2 over horizon 1 gives some touch or survival risk. What barrier over horizon 4 preserves that same standardized ratio?随机过程中等derivation未尝试面试订阅3524Barrier Matching a Unit-Horizon Risk Across Horizon 9A barrier of 1 over horizon 1 gives some touch or survival risk. What barrier over horizon 9 preserves that same standardized ratio?随机过程中等derivation未尝试面试订阅3525Joint Max-End Probability With t = 1, a = 1, b = 0For standard Brownian motion, what is P(max 0<=s<=1 W s >= 1, W 1 <= 0)?随机过程中等derivation未尝试面试订阅3528Barrier Needed for a 5% Joint Tail With Endpoint Cap 0For standard Brownian motion over t = 1, what barrier a gives P(max 0<=s<=1 W s >= a, W 1 <= 0) = 0.05?随机过程困难derivation未尝试面试订阅3529Endpoint Cap Needed for a 10% Joint TailFor standard Brownian motion over t = 1 with barrier a = 1, what endpoint cap b gives P(max 0<=s<=1 W s >= 1, W 1 <= b) = 0.10?随机过程困难derivation未尝试面试订阅3530Barrier Survival Probability for a = 1 Over t = 1For standard Brownian motion, what is P(max 0<=s<=1 W s < 1)?随机过程困难derivation未尝试面试订阅3532Barrier Survival Probability for a = 2 Over t = 0.5For standard Brownian motion, what is P(max 0<=s<=0.5 W s < 2)?随机过程中等derivation未尝试面试订阅3533Barrier Needed for 90% Survival Over One Time UnitFor standard Brownian motion over t = 1, what barrier a gives P(max 0<=s<=1 W s < a) = 0.90?随机过程中等derivation未尝试面试订阅3534Horizon Giving 80% Survival Under Barrier 1.5For standard Brownian motion, what horizon t gives P(max 0<=s<=t W s < 1.5) = 0.80?随机过程中等derivation未尝试面试订阅3535Ratio of Touch Probabilities for Barriers 1 and 1.5Over horizon t = 1 for standard Brownian motion, what is P(M 1 >= 1) / P(M 1 >= 1.5)?随机过程中等derivation未尝试面试订阅3536Why Reflection Solves a Path Event Using an Endpoint TailWhy does the reflection principle let a maximum-over-time event collapse to a Gaussian endpoint tail?随机过程中等essay未尝试面试订阅3537Finite-Horizon Touching Is Not the Same as Eventual HittingWhy is a finite-horizon barrier-touch probability conceptually different from the statement that Brownian motion eventually hits every level?随机过程中等essay未尝试面试订阅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