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544Hitting Time on the Diamond GraphTake the complete graph K 4 on vertices \ A, B, C, D\ and remove edge A - D, leaving 5 edges (the "diamond" or "kite" graph). The resulting degrees are d(A) = d(D) = 2 and d(B) = d(C) = 3. A simple random walk moves at each step to a uniformly random neighbor. (a) Starting from vertex A, find the expected hitting time h(A D). (b) Starting from vertex D, find h(D A). (c) Compute the commute time C(A, D) and verify it using the effective-resistance formula.概率困难数值题未尝试面试订阅548Commute Time Between Endpoints of a PathConsider the path graph P n with vertices \ 0, 1, \ldots, n-1\ and n-1 edges (each of unit resistance), where the walk at interior vertices moves left or right with equal probability, and at the endpoints moves to the unique neighbor. (a) Compute the effective resistance R eff (0, n-1) between the two endpoints. (b) Using C(u,v) = 2m R eff (u,v), find the commute time between the two endpoints. (c) Verify for n = 4 by computing h(0 3) and h(3 0) directly.概率中等derivation未尝试免费550Expected Cover Time of the Cycle C₆A simple random walk moves on the cycle graph C 6 (vertices 0, 1, \ldots, 5). At each step, the walker moves clockwise or counterclockwise with equal probability. Starting at vertex 0, what is the expected number of steps to visit all 6 vertices (the expected cover time)?概率困难derivation未尝试面试订阅551Expected Occupied Labels 110 independent packets are hashed uniformly into 8 labels. What is the expected number of labels hit at least once?概率简单数值题未尝试免费556Expected Singleton Labels 18 independent arrivals are assigned uniformly to 10 labels. What is the expected number of labels that receive exactly one arrival?概率简单数值题未尝试免费576Discrete Signal Stop Rule 1You may inspect up to 2 independent signals, each uniformly distributed on 1,...,7 . After seeing a signal, you may lock it in and stop. Rejecting a signal and continuing costs 1 point(s). If you reach the last draw, you must take it. What first-round acceptance threshold is optimal, and what is the expected net score?概率简单数值题未尝试免费586Weighted Offer Stop Rule 1You may inspect up to 2 independent offers. Each offer takes values 1 with probability 1/4, 4 with probability 1/2, 9 with probability 1/4. Rejecting an offer and continuing costs 1 point(s), and if you reach the last draw you must accept it. What first-round acceptance threshold is optimal, and what is the resulting expected net payoff?概率简单数值题未尝试免费597Continuation Value Calibration 2A trader may inspect up to 3 independent candidate fills with support [1, 4, 8, 11] and probabilities ['1/4', '1/4', '1/4', '1/4']. Rejecting a fill and continuing costs 1. What first observation becomes just good enough to accept, and what is the overall optimal expected net value?概率简单数值题未尝试免费606Target-Hitting Stake Choice 6You start with wealth 5. In each of at most 3 rounds, you may bet any integer stake between 0 and your current wealth on an even-money coin that wins with probability 3/5. If you win, your wealth increases by the stake; if you lose, it decreases by the stake. What first-round stake maximizes the probability of finishing with wealth at least 9 after 3 rounds, and what is that maximal probability?概率简单数值题未尝试免费1426Cubic Residue of an Exponential RemainderCompute lim x->0 [e (3x) - 1 - 3x - (9/2)x 2] / x 3.数学简单数值题未尝试免费1428Exponential Quadratic Remainder 3Compute lim (x->0) [exp(3x + -2x 2) - 1 - 3x] / x 2.数学中等数值题未尝试免费1429Square-Root Cubic CoefficientCompute lim x->0 [sqrt(1+4x) - 1 - 2x + 2x 2] / x 3.数学中等数值题未尝试免费1430Fifth-Order Trigonometric ResidueCompute lim x->0 [sin(3x) - 3x + (9/2)x 3] / x 5.数学困难数值题未尝试免费1431Half-Power Expansion With Cubic TermCompute lim x->0 [(1+5x) (1/2) - 1 - (5/2)x + (25/8)x 2] / x 3.数学简单数值题未尝试免费1432Rational Blow-Up CancellationCompute lim x->0 [(1-2x) (-1) - 1 - 2x] / x 2.数学中等数值题未尝试免费1433Fractional Linear RemainderCompute lim x->0 [((1+x)/(1-x)) - 1 - 2x] / x 2.数学中等数值题未尝试免费1434Tangent Minus IdentityCompute lim x->0 (tan x - x) / x 3.数学中等数值题未尝试免费1435Logarithmic Mixed Expansion 5Compute lim (x->0) [ln(1 + 5x + 2x 2) - 5x] / x 2.数学困难数值题未尝试面试订阅1437Cube-Root LinearizationCompute lim x->0 [ (1+2x) (1/3) - 1 ] / x.数学简单数值题未尝试免费1438Sequence Limit From a Square RootCompute lim n->∞ n[(1+2/n) (1/2) - 1].数学中等数值题未尝试免费