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4970Probability of Either No Jump by Time 1 or a First Jump to CFrom state A, the jump rates to B and C are 0.4 and 0.6. What is the probability that by time 1 either no jump has happened yet or, if a jump has happened, the first jump was to C?随机过程简单数值题未尝试面试订阅4971Expected Time to Absorption With One Reset LinkA CTMC has states 0, 1, 2 with state 2 absorbing. From 0 it jumps to 1 at rate a=1, and from 1 it jumps to 2 at rate b=2 or back to 0 at rate c=1. What is the expected time to hit state 2 starting from 0?随机过程中等数值题未尝试面试订阅4972Reset Rate Implied by a Target Expected Hitting TimeA CTMC has states 0,1,2 with state 2 absorbing. From 0 it jumps to 1 at rate a=1, and from 1 it jumps to 2 at rate b=1.5 or back to 0 at rate c. If the expected time to hit state 2 from state 0 is 3, what is c?随机过程中等数值题未尝试面试订阅4973Forward Rate Implied by a Target Expected Hitting TimeA CTMC has states 0,1,2 with state 2 absorbing. The rates are a=2 from 0 to 1 and c=1 from 1 back to 0, while the rate from 1 to 2 is b. If the expected time to hit state 2 from state 0 is 1.5, what is b?随机过程中等数值题未尝试面试订阅4974Reduction in Expected Hitting Time After Lowering the Reset RateIn the same 0→1→2 with reset model, let a=1 and b=2. If the return rate c drops from 3 to 1, by how much does the expected hitting time from 0 to 2 fall?随机过程中等数值题未尝试面试订阅4975Stationary Distribution of a Three-State Birth-Death ChainA three-state birth-death CTMC has rates 0->1 = 1, 1->0 = 2, 1->2 = 3, and 2->1 = 1.5. What is its stationary distribution (pi 0, pi 1, pi 2)?随机过程中等数值题未尝试面试订阅4976Infer Missing Birth-Death Rate From Stationarity 11A three-state birth-death CTMC has stationary distribution (0.5, 0.3, 0.2). The rates 0->1 and 1->2 are 0.6 and 0.4, and the rate 1->0 is 1. What rate 2->1 is needed for stationarity?随机过程困难数值题未尝试面试订阅4977Stationary Ratio Between State 2 and State 0A three-state birth-death CTMC has rates 0->1 = 0.8, 1->0 = 0.4, 1->2 = 0.6, and 2->1 = 1.2. What is the stationary ratio pi 2 / pi 0?随机过程困难数值题未尝试面试订阅4978Missing Return Rate for a Uniform Stationary DistributionA three-state birth-death CTMC has rates 0->1 = 1, 1->0 = 1, and 1->2 = 0.8. What rate 2->1 makes the stationary distribution uniform across the three states?随机过程困难数值题未尝试面试订阅4979Infer Missing Birth-Death Rate From Stationarity 14A three-state birth-death CTMC has stationary distribution (0.2, 0.5, 0.3). The rates 0->1 and 1->2 are 1.5 and 0.9, and the rate 1->0 is 0.6. What rate 2->1 is implied?随机过程困难数值题未尝试面试订阅4980Short-Horizon Expected State Reward 1At state i, jump rates to j and k are 0.4 and 0.6. Let the state rewards be V(i)=0, V(j)=5, and V(k)=-2. Using a first-order CTMC approximation over Delta t = 0.1, what is E[V(X Delta t ) | X 0=i]?随机过程困难数值题未尝试面试订阅4981Expected Hitting Time 1A CTMC has states 0,1,2 with state 2 absorbing. From 0 it jumps to 1 at rate 0.8. From 1 it jumps to 2 at rate 1.2 and back to 0 at rate 0.4. What is the expected time to hit state 2 starting from state 0?随机过程困难数值题未尝试面试订阅4982Short-Horizon Expected State Reward 2A CTMC starts in state i. Over a short interval Delta t = 0.05, the jump rates from i to j, k, l are 1, 0.5, and 0.5. State rewards are V(i)=1, V(j)=10, V(k)=4, and V(l)=-6. What is the first-order approximation to E[V(X Delta t )]?随机过程困难数值题未尝试面试订阅4983Infer Return Rate From Expected Hitting Time 17A CTMC has states 0,1,2 with state 2 absorbing. From 0 it jumps to 1 at rate a=0.5. From 1 it jumps to 2 at rate b=1 and back to 0 at rate c. If the expected time to hit 2 from 0 is 4, what is c?随机过程困难数值题未尝试面试订阅4984Short-Horizon Probability of Landing in a SubsetA CTMC starts in state i. The jump rates from i to j, k, l are 0.7, 0.2, and 0.1. Using a first-order approximation over Delta t = 0.2, what is the probability that X Delta t lies in the subset j,k ?随机过程困难数值题未尝试面试订阅4985Infer Return Rate From Expected Hitting Time 19A CTMC has states 0,1,2 with state 2 absorbing. From 0 to 1 the rate is a=1.5, and from 1 to 2 the rate is b=0.5. If the expected hitting time of state 2 from 0 is 3, what is the return rate c from 1 back to 0?随机过程困难数值题未尝试面试订阅4986Why Fixed Waiting Times Break the CTMC PropertyA simulator gets the jump-chain routing probabilities right but replaces exponential waits by deterministic one-minute waits in every state. Why is the resulting calendar-time process generally not a CTMC anymore?随机过程困难essay未尝试面试订阅4987Why Uniformization Can Use One Poisson ClockWhy can a CTMC with different exit rates across states still be simulated using one common Poisson clock plus occasional virtual self-jumps?随机过程困难essay未尝试面试订阅4988Stationary Does Not Mean SlowWhy can a state have a small stationary probability even if it has a large exit rate?随机过程困难essay未尝试面试订阅4989Same Jump Chain, Different Calendar-Time BehaviorWhy can two jump processes share exactly the same jump chain but still look very different when observed in real time?随机过程困难essay未尝试面试订阅