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3662Why a Stochastic Integral Is the Default Martingale CandidateWhy are square-integrable stochastic integrals usually the first objects you test as martingales in continuous time?随机过程中等essay未尝试面试订阅3663Why Compensation Terms Keep ReappearingWhy do compensation terms like -t or - frac12 a 2 t appear so often when building martingales from Brownian motion?随机过程中等essay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3671Expected Exit Time from [0,5] Starting at 2Brownian motion starts at 2 and stops on first exit from [0,5]. Using OST on W t 2-t, what is E[tau]?随机过程中等derivation未尝试面试订阅3672Expected Exit Time from [-2,4] Starting at 1Brownian motion starts at 1 and stops on first exit from [-2,4]. Using OST on W t 2-t, what is E[tau]?随机过程中等derivation未尝试面试订阅3673Symmetric Half-Width for Mean Exit Time 12Brownian motion starts at 2 inside [-L,L]. Using OST on W t 2-t, what L makes E[tau] = 12?随机过程中等derivation未尝试面试订阅3674Upper Barrier for Mean Exit Time 6 from x=1.5Brownian motion starts at 1.5 with lower barrier 0 and upper barrier b. Using OST on W t 2-t, what b makes E[tau] = 6?随机过程中等derivation未尝试面试订阅3675Average Exit Time with Uniform Start on [0,6]The starting point X is uniform on [0,6]. Brownian motion starts at X and stops on first exit from [0,6]. What is E[tau] on average?随机过程中等derivation未尝试面试订阅3676Drifted Brownian Upper-Hit Probability with Positive DriftA drifted Brownian motion satisfies dX t = 0.5dt + dW t, starts at 1, and stops on first exit from [0,4]. Using the exponential martingale, what is the probability it exits through 4?随机过程中等derivation未尝试面试订阅3677Drifted Brownian Upper-Hit Probability with Negative DriftA drifted Brownian motion satisfies dX t = -0.3dt + dW t, starts at 2, and stops on first exit from [0,5]. What is the probability it exits through 5?随机过程中等derivation未尝试面试订阅3678Start Needed for a 60% Drifted Upper-Hit ProbabilityA drifted Brownian motion satisfies dX t = 0.4dt + dW t and stops on first exit from [0,6]. For what start x does the upper-exit probability equal 0.6?随机过程中等derivation未尝试面试订阅3679Upper Barrier Needed for an 85% Drifted Hit ProbabilityA drifted Brownian motion satisfies dX t = 0.7dt + dW t, starts at 1, and stops on first exit from [0,b]. What upper barrier b makes the upper-exit probability equal 0.85?随机过程中等derivation未尝试面试订阅3680Drifted Brownian Upper-Hit Probability on a Short IntervalA drifted Brownian motion satisfies dX t = 0.2dt + dW t, starts at 2, and stops on first exit from [0,3]. What is the upper-exit probability?随机过程中等derivation未尝试面试订阅3681Upper-Hit Probability with Drift and Nonunit VolatilityA diffusion satisfies dX t = 0.4dt + 1.5dW t, starts at 1.5, and stops on first exit from [0,5]. What is the probability it exits through 5?随机过程中等derivation未尝试面试订阅3682Upper-Hit Probability with Negative Drift and Lower VolatilityA diffusion satisfies dX t = -0.2dt + 0.8dW t, starts at 1, and stops on first exit from [0,4]. What is the upper-exit probability?随机过程中等derivation未尝试面试订阅3683Start Needed for a Drifted Hit Probability with Nonunit VolatilityA diffusion satisfies dX t = 0.3dt + 1.2dW t and stops on first exit from [0,6]. For what start x does the upper-exit probability equal 0.65?随机过程中等derivation未尝试面试订阅