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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3566Cubic Brownian Polynomial MartingaleChoose a so that M t = W t 3 + a t W t is a martingale.随机过程中等derivation未尝试面试订阅3567Quartic Brownian Polynomial MartingaleChoose a and b so that M t = W t 4 + a t W t 2 + b t 2 is a martingale.随机过程中等derivation未尝试面试订阅3568Quintic Brownian Polynomial MartingaleChoose a and b so that M t = W t 5 + a t W t 3 + b t 2 W t is a martingale.随机过程中等derivation未尝试面试订阅3569Exponential-Cosine Local MartingaleChoose a so that M t = e a t cos(2W t) is a local martingale.随机过程中等derivation未尝试面试订阅3570Exponential-Sine Local MartingaleChoose a so that M t = e a t sin(3W t) is a local martingale.随机过程中等derivation未尝试面试订阅3571Nonzero Power Making S_t^p DriftlessA GBM satisfies dS t = 0.04 S t dt + 0.2 S t dW t. Find the nonzero p such that S t p is a local martingale.随机过程中等derivation未尝试面试订阅3572Discount Rate Making e^{-gamma t} S_t^2 DriftlessA GBM satisfies dS t = 0.03 S t dt + 0.2 S t dW t. What gamma makes e -gamma t S t 2 a local martingale?随机过程中等derivation未尝试面试订阅3573Time Shift Making log S_t DriftlessA GBM satisfies dS t = 0.07 S t dt + 0.3 S t dW t. What c makes log S t + c t a local martingale?随机过程中等derivation未尝试面试订阅3574Dynamics of the Reciprocal of GBMA GBM satisfies dS t = 0.05 S t dt + 0.3 S t dW t. If Y t = 1 / S t, what are the drift and diffusion coefficients of Y t?随机过程中等derivation未尝试面试订阅3575Dynamics of the Square Root of GBMA GBM satisfies dS t = 0.02 S t dt + 0.4 S t dW t. If Z t = S t 1/2 , what are the drift and diffusion coefficients of Z t?随机过程中等derivation未尝试面试订阅3576Reciprocal Transform That Removes Drift CompletelyA diffusion satisfies dX t = X t dt + X t dW t. If Y t = 1 / X t, what SDE does Y t satisfy?随机过程中等derivation未尝试面试订阅3577Integrating Factor That Linearizes an Affine DiffusionA diffusion satisfies dX t = (2X t + 1) dt + 3 dW t. Define Y t = e -2t (X t + 1/2). What SDE does Y t satisfy?随机过程中等derivation未尝试面试订阅