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491Hitting Time with State-Dependent Self-LoopsA Markov chain on \ 0, 1, 2, 3\ has transition probabilities: from state i (0 \le i \le 2), the chain stays at i with probability i 4 , moves to i+1 with probability 1 - i 4 . State 3 is absorbing. Compute E[T 3 \mid X 0 = 0].概率简单数值题未尝试免费496First Passage in a Three-State Chain with Self-LoopA Markov chain on \ 0, 1, 2\ has transition matrix P = \begin pmatrix 1 & 0 & 0 \\ \tfrac 1 5 & \tfrac 2 5 & \tfrac 2 5 \\ 0 & \tfrac 3 4 & \tfrac 1 4 \end pmatrix . State 0 is absorbing. Compute E[T 0 \mid X 0 = 1], where T 0 = \inf\ n \ge 1 : X n = 0\ .概率简单数值题未尝试免费497Hitting Time with Harmonically Increasing Forward ProbabilityA Markov chain on \ 0, 1, 2, 3\ has transitions: from state i (0 \le i \le 2), move to i+1 with probability i+1 i+2 and stay at i with probability 1 i+2 . State 3 is absorbing. Compute E[T 3 \mid X 0 = 0].概率简单derivation未尝试免费501Biased Gambler's Ruin ProbabilityA gambler starts with \3 and plays a sequence of independent rounds. Each round she wins \1 with probability p = 0.4 and loses \1 with probability q = 0.6. She stops when her fortune reaches \0 (ruin) or \8. What is the probability that she is ruined?概率简单数值题未尝试免费526Hitting Time on the Complete Graph K₅A random walk moves on the complete graph K 5 (five vertices, every pair connected). At each step, the walker moves to one of the 4 neighbors chosen uniformly at random. Starting from vertex u, what is the expected number of steps to reach a specified vertex v u for the first time?概率简单数值题未尝试免费527Hitting Time to the Antipodal Vertex on a CycleA simple random walk moves on the cycle graph C 8 (vertices 0, 1, \ldots, 7 arranged in a circle). At each step, the walker moves clockwise or counterclockwise with equal probability \tfrac 1 2 . Starting at vertex 0, what is the expected number of steps to reach the antipodal vertex 4 for the first time?概率简单数值题未尝试免费531Cover Time of the Path Graph P₃A random walk moves on the path graph P 3 with vertices \ 1, 2, 3\ and edges \ 1 - 2, 2 - 3\ . At each step, the walker moves to a uniformly random neighbor (so from vertex 2 it goes to 1 or 3 each with probability \tfrac 1 2 , and from vertex 1 or 3 it moves deterministically to 2). Starting at vertex 1, what is the expected number of steps to visit all three vertices (the cover time)?概率简单数值题未尝试免费534Hitting Time from Leaf to Root on a Complete Ternary TreeConsider the complete ternary tree of depth 2: a root vertex r with 3 children, each of which has 3 children (leaves), giving 13 vertices total. A simple random walk moves at each step to a uniformly random neighbor. Starting from a leaf vertex, what is the expected number of steps to reach the root r for the first time?概率中等数值题未尝试免费536Expected Return Time on the Complete Graph K₄A simple random walk moves on the complete graph K 4 (four vertices, every pair connected). At each step, the walker moves to one of the 3 neighbors chosen uniformly at random. Starting at a vertex v, what is the expected number of steps to return to v for the first time?概率简单数值题未尝试免费537Hitting Time Between Leaves on the Star Graph S₅The star graph S 5 has a central hub vertex c connected to 4 leaf vertices \ \ell 1, \ell 2, \ell 3, \ell 4\ . A simple random walk at each step moves to a uniformly random neighbor: from the hub, it goes to each leaf with probability \tfrac 1 4 ; from any leaf, it goes to c with probability 1. Starting at leaf \ell 1, what is the expected number of steps to reach leaf \ell 2 for the first time?概率简单数值题未尝试免费541Stationary Distribution and Return Times on a Small GraphConsider the graph G on four vertices \ A, B, C, D\ with edges \ A - B,\, A - C,\, A - D,\, B - C\ , so the degree sequence is d(A)=3, d(B)=2, d(C)=2, d(D)=1. A simple random walk moves at each step to a uniformly random neighbor. (a) Find the stationary distribution . (b) Compute the expected return time E v[T v +] for each vertex v.概率简单数值题未尝试免费542Hitting Time on the Wheel Graph W₆The wheel graph W 6 consists of a central hub h connected to all 5 vertices of a cycle C 5 (so h has degree 5 and each rim vertex has degree 3: two cycle neighbors and the hub). A simple random walk moves at each step to a uniformly random neighbor. Starting from a rim vertex v, what is the expected number of steps to reach the hub h?概率中等数值题未尝试免费546Hitting Time on the Path Graph P₅A simple random walk moves on the path graph P 5 with vertices \ 0,1,2,3,4\ and edges connecting consecutive vertices. At the interior vertices (1, 2, 3), the walker moves left or right with equal probability \tfrac 1 2 . At the endpoints (0 and 4), the walker moves to the unique neighbor with probability 1. Starting at vertex 0, what is the expected number of steps to reach vertex 4 for the first time?概率简单数值题未尝试免费547Hitting Time on K₄ Minus One EdgeTake the complete graph K 4 on vertices \ 1,2,3,4\ and remove edge \ 1,4\ . The resulting graph has 5 edges, with d(1)=d(4)=2 and d(2)=d(3)=3. A simple random walk moves at each step to a uniformly random neighbor. Starting from vertex 2, what is the expected number of steps to reach vertex 4?概率简单数值题未尝试免费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561Tagged Label Survives Unseen 1A particular tag is one of 9 equally likely outcomes on each of 11 independent draws. What is the probability that this tag is never drawn?概率简单数值题未尝试免费566Time to Reach 3 Distinct Types 1Coupons arrive uniformly from 7 types. What is the expected number of draws needed to see 3 distinct types for the first time?概率简单数值题未尝试免费571All Labels Seen by Draw 5 1Draw 5 coupons independently and uniformly from 3 types. What is the probability that all 3 types have appeared at least once by time 5?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