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4915Spectral Weight Design 25Why should a spectral risk measure place weakly larger weights on worse outcomes than on mild ones?数理金融困难essay未尝试面试订阅4961Shooting Method Diagnostic 18Why does the shooting method for a second-order BVP usually search over one missing initial slope rather than over an entire function?数学困难essay未尝试面试订阅4962Multiplicity Diagnostic 19Why can a boundary value problem have multiple solutions even when the corresponding initial-value problem is unique for each chosen slope?数学困难essay未尝试面试订阅4963Neumann Compatibility Diagnostic 20Why does a pure Neumann problem for y'' = f sometimes need a compatibility condition before any solution can exist?数学困难essay未尝试面试订阅4964Finite-Difference Diagnostic 21Why can a finite-difference solve be more numerically stable than naive shooting for a stiff BVP?数学困难essay未尝试面试订阅4966Probability the First Observed Jump by Time 0.5 Is to BFrom state A, the jump rates to B and C are 0.8 and 1.2. What is the probability that by time 0.5 a jump has occurred and that first jump was to B?随机过程简单数值题未尝试面试订阅4967Probability the First Observed Jump by Time 2 Is to CFrom state A, the jump rates to B, C, and D are 0.3, 0.5, and 0.2. What is the probability that by time 2 the first jump has occurred and it was to C?随机过程简单数值题未尝试面试订阅4968Probability of No Jump by Time 0.4From state A, the jump rates to B and C are 1.5 and 0.5. What is the probability that no jump has occurred by time 0.4?随机过程简单数值题未尝试面试订阅4969Probability the First Observed Jump by Time 1 Is to DFrom state A, the jump rates to B, C, and D are 0.6, 0.9, and 0.5. What is the probability that by time 1 the first jump has occurred and it was to D?随机过程简单数值题未尝试面试订阅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