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495Finite Hitting Time Between Communicating ComponentsConsider a Markov chain on \ 1, 2, 3, 4, 5\ with transition matrix P = \begin pmatrix \tfrac 1 2 & \tfrac 1 2 & 0 & 0 & 0 \\ \tfrac 1 3 & 0 & \tfrac 2 3 & 0 & 0 \\ 0 & \tfrac 1 4 & 0 & \tfrac 1 2 & \tfrac 1 4 \\ 0 & 0 & 0 & \tfrac 1 2 & \tfrac 1 2 \\ 0 & 0 & 0 & \tfrac 1 3 & \tfrac 2 3 \end pmatrix . Let B = \ 4, 5\ and T B = \inf\ n \ge 0 : X n \in B\ . **(a)** Show that E[T B \mid X 0 = i] < for all i \in \ 1, 2, 3\ . **(b)** Compute E[T B \mid X 0 = 1].概率困难derivation未尝试免费500Harmonic Function Method for Splitting Probability and Hitting TimeA Markov chain on \ 0, 1, 2, 3, 4, 5\ has absorbing states 0 and 5. For interior states: p(1,0) = \tfrac 1 4 , \quad p(1,2) = \tfrac 3 4 , p(2,1) = \tfrac 1 3 , \quad p(2,3) = \tfrac 2 3 , p(3,2) = \tfrac 1 2 , \quad p(3,4) = \tfrac 1 2 , p(4,3) = \tfrac 2 3 , \quad p(4,5) = \tfrac 1 3 . Let T = \inf\ n \ge 0 : X n \in \ 0, 5\ \ . **(a)** A function f on \ 0,\ldots,5\ is *harmonic* on \ 1,2,3,4\ if f(i) = \sum j p(i,j) f(j) for i \in \ 1,2,3,4\ . Find the harmonic function with f(0) = 0, f(5) = 1, and use it to compute P(X T = 5 \mid X 0 = 2). **(b)** Compute E[T \mid X 0 = 2].概率困难derivation未尝试免费503Asymmetric Step-Size Gambler's RuinA gambler starts with \3. Each round she wins \1 with probability \tfrac 2 3 or loses \2 with probability \tfrac 1 3 . She stops when her fortune first hits \0 or below (ruin) or reaches \5 or above (victory). Note that if her fortune is \1 and she loses, she goes to -\1, which counts as ruin. What is the probability that she achieves victory?概率中等数值题未尝试免费504Three-Player Elimination GameThree players A, B, C hold tokens: A has 2, B has 1, C has 1 (total 4). Each round, two players are chosen uniformly at random (from the \binom 3 2 = 3 pairs) and they play a fair game: each of the two has probability \tfrac 1 2 of winning one token from the other. A player with 0 tokens is eliminated. Play continues until one player holds all 4 tokens. What is the probability that player A (who starts with 2 tokens) ends up with all 4?概率中等数值题未尝试免费505Gambler's Ruin with a Partially Reflecting BarrierConsider a Markov chain on \ 0, 1, 2, \ldots\ with absorbing state 0. From state k \ge 1, the chain moves to k+1 with probability p and to k-1 with probability q = 1 - p, where 0 < p < 1. However, there is a **reflecting barrier** at state N: from state N, the chain moves to N-1 with probability 1 (it is always pushed back). Starting from state k with 1 \le k \le N: **(a)** Find the probability r k of eventual absorption at 0. **(b)** For p = q = \tfrac 1 2 and N = 4, compute r 2 and the expected absorption time E[T \mid X 0 = 2].概率困难derivation未尝试免费529Hitting Time on the Complete Bipartite Graph K₃,₃Consider the complete bipartite graph K 3,3 with parts A = \ a 1, a 2, a 3\ and B = \ b 1, b 2, b 3\ , where every vertex in A is connected to every vertex in B and vice versa (no edges within a part). A random walk at any vertex moves to each of its 3 neighbors with equal probability \tfrac 1 3 . Starting from vertex a 1, what is the expected number of steps to reach vertex b 1 for the first time?概率中等数值题未尝试免费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543Hitting Time on the Ladder Graph (2×3 Grid)Consider the 2 3 grid graph (ladder graph) with vertices arranged as: \begin matrix 1 & - & 2 & - & 3 \\ | & & | & & | \\ 4 & - & 5 & - & 6 \end matrix Edges connect horizontal and vertical neighbors. A simple random walk moves at each step to a uniformly random neighbor. Starting at corner vertex 1 (degree 2), what is the expected number of steps to reach the opposite corner vertex 6 (degree 2)?概率中等数值题未尝试免费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1544Sherman-Morrison Vector Update D 4Suppose A is invertible, A (-1)b = [1, 3] T, A (-1)u = [0, 2] T, v T A (-1)b = 4, and v T A (-1)u = 2/3. What is (A + u v T) (-1) b?数学困难数值题未尝试面试订阅1605Hedged Pair Variance 4Two assets have variances 1 and 9, with covariance -1. What is the variance of the portfolio with weights (1/3, 2/3)?数学困难数值题未尝试面试订阅1608Minimum-Variance Hedge Ratio 1You hedge X with h units of Y and want to minimize Var(X - hY). If Cov(X, Y) = 8 and Var(Y) = 4, what is the optimal hedge ratio h?数学中等derivation未尝试免费1618Negative Correlation Extraction 2Two assets have variances 4 and 25, and covariance -6. What is their correlation?数学中等derivation未尝试免费1628Estimating a Zero-Inflated Order-Arrival ModelPer-second order arrivals are modeled as follows: with probability the market is inactive and the observed count is exactly 0; with probability 1- , the count is Poisson ( ). From data, the empirical zero frequency is 0.70 and the empirical mean count is 0.60. Use the method of moments to estimate ( , ).统计困难derivation未尝试面试订阅1630Shifted Exponential Calibration from Raw MomentsA toy latency model assumes X = c + Y, \qquad Y \sim Exp ( ), with unknown deterministic floor c>0 and unknown rate . From historical data, the empirical mean of X is 8 and the empirical second raw moment is 73. Use the method of moments to estimate c and .统计简单derivation未尝试免费