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3635Coefficient Making [aW_t-B_t, W_t+B_t]_2 Equal 6Choose a so that [aW t - B t, W t + B t] 2 = 6 with independent W and B.随机过程中等derivation未尝试面试订阅3636Why Smooth Drift Terms Disappear from Quadratic VariationWhy do smooth finite-variation terms fail to contribute to quadratic variation even though they can dominate the path in level?随机过程中等essay未尝试面试订阅3637Why Independent Brownian Motions Have Zero CovariationWhy does independence of Brownian drivers force the covariation term to vanish?随机过程中等essay未尝试面试订阅3638Why Quadratic Variation Scales with the Square of a CoefficientWhy does multiplying a Brownian-driven process by c multiply its quadratic variation by c 2 rather than by c?随机过程中等essay未尝试面试订阅3639Why Quadratic Variation Identifies Diffusion StrengthWhy is quadratic variation often the cleanest object for reading off the local diffusion strength of a process?随机过程中等essay未尝试面试订阅3640Why a Time Change Alters Quadratic VariationWhy does replacing W t by W ct change quadratic variation even before any extra scaling coefficient is added?随机过程中等essay未尝试面试订阅3641Coefficient Making W_t^2-a t a MartingaleChoose a so that X t = W t 2 - a t is a martingale.随机过程中等derivation未尝试面试订阅3642Coefficient Making exp(aW_t-2t) a MartingaleChoose a so that X t = exp(aW t - 2t) is a martingale.随机过程中等derivation未尝试面试订阅3643Coefficient Making e^{a t} cos W_t a MartingaleChoose a so that X t = e a t cos W t is a martingale.随机过程中等derivation未尝试面试订阅3653Choose c for a Cubic Brownian Local MartingaleFor standard Brownian motion, choose c so that X t=W t 3-c\int 0 t W s\,ds is a local martingale.随机过程中等derivation未尝试面试订阅3655Choose c in the Quartic Brownian PolynomialFor standard Brownian motion, choose c so that X t=W t 4-6tW t 2+ct 2 is a martingale.随机过程中等derivation未尝试面试订阅3656Classify an Exponentially Weighted Stochastic IntegralClassify X t=\int 0 t e -s \,dW s as a martingale, submartingale, or supermartingale.随机过程中等derivation未尝试面试订阅3657Classify a Purely Increasing Adapted ProcessClassify X t=\int 0 t e -s \,ds as a martingale, submartingale, or supermartingale.随机过程中等derivation未尝试面试订阅3658Brownian Motion Plus Accumulated Variance BudgetClassify X t=W t+\int 0 t W s 2\,ds as a martingale, submartingale, or supermartingale.随机过程中等derivation未尝试面试订阅3660Stochastic Integral with a Positive Deterministic CushionClassify X t=\int 0 t W s\,dW s+0.2t as a martingale, submartingale, or supermartingale.随机过程中等derivation未尝试面试订阅3666Upper-Hit Probability on [-3,5] from x=1A Brownian motion starts at x = 1 and is stopped at the first exit from [-3,5]. Using OST on W t, what is the probability it exits through 5?随机过程中等derivation未尝试面试订阅3667Start Required for a 70% Upper Exit Chance on [0,6]Brownian motion starts at x inside [0,6] and stops on first exit. For what x does OST imply a 70% chance of exiting through 6?随机过程中等derivation未尝试面试订阅3668Lower Barrier for a 40% Upper Exit ChanceBrownian motion starts at 1 with upper barrier 4 and lower barrier a. Using OST on W t, what a gives a 40% chance of exiting through 4?随机过程中等derivation未尝试面试订阅3669Symmetric Half-Width for a 75% Upper Exit ChanceBrownian motion starts at 2 inside [-L,L]. Using OST on W t, what L gives a 75% chance of hitting +L before -L?随机过程中等derivation未尝试面试订阅3670Average Upper-Exit Probability with Uniform Start on [0,8]The starting point X is uniform on [0,8]. Brownian motion starts at X and stops on first exit from [0,8]. What is the average probability of exiting through 8?随机过程中等derivation未尝试面试订阅