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2812Expected Squared Distance in the Unit SquareTwo points are chosen independently and uniformly in the unit square. What is the expected squared Euclidean distance between them?脑筋急转弯中等derivation未尝试面试订阅2813Largest Barycentric Piece Exceeds HalfA point is chosen uniformly inside a triangle. Joining the point to the three vertices partitions the triangle into three smaller triangles. What is the probability that the largest of those three area fractions exceeds 1/2?脑筋急转弯中等derivation未尝试面试订阅2814Closer to the Center Than to the OriginA point is chosen uniformly in the unit square. What is the probability that it is closer to the center (1/2,1/2) than to the origin (0,0)?脑筋急转弯中等derivation未尝试面试订阅2891Two-State Regime SwitchingA market regime is either Calm or Volatile. From Calm, the chain moves to Volatile with probability ; from Volatile, it moves to Calm with probability . Find the stationary distribution.概率简单derivation未尝试面试订阅2892A Three-State Birth-Death Regime ChainConsider states Bull, Neutral, and Bear with transition matrix \[ P= \begin pmatrix 0.8 & 0.2 & 0\\ 0.3 & 0.4 & 0.3\\ 0 & 0.2 & 0.8 \end pmatrix . \] Find the stationary distribution.概率中等derivation未尝试面试订阅2893Uniform Stationarity for a Doubly Stochastic ChainSuppose a finite Markov chain has transition matrix P whose rows and columns both sum to 1. Show that the uniform distribution is stationary.概率简单derivation未尝试面试订阅2894Random Walk on an Undirected GraphA simple random walk moves on a connected undirected graph G=(V,E) by choosing a uniformly random neighbor at each step. Show that the stationary distribution is proportional to degree.概率简单derivation未尝试面试订阅2895Star Graph Long-Run OccupancyA simple random walk runs on a star graph with one hub and m leaves. What fraction of time is spent at the hub in stationarity, and what fraction at each leaf?概率简单derivation未尝试面试订阅2896Complete Bipartite Graph K_{2,3}A simple random walk runs on the complete bipartite graph K 2,3 . What is the stationary probability of each vertex, and what total stationary mass sits on each side of the bipartition?概率中等derivation未尝试面试订阅2897Triangle with a Pendant NodeConsider a simple random walk on the graph with vertices A,B,C,D, where A,B,C form a triangle and D is connected only to A. Find the stationary distribution.概率中等derivation未尝试面试订阅2898Directed Cycle with Clockwise DriftOn the directed cycle with states \ 0,1,\dots,n-1\ , from each state the chain moves clockwise with probability p and counterclockwise with probability 1-p. Show that the stationary distribution is uniform.概率中等derivation未尝试面试订阅2899Reflecting Simple Walk on 0 to mConsider the chain on \ 0,1,\dots,m\ . From an interior state it moves left or right with probability 1/2 each. From 0 it stays put with probability 1/2 and moves to 1 with probability 1/2; from m it stays put with probability 1/2 and moves to m-1 with probability 1/2. Find the stationary distribution.概率中等derivation未尝试面试订阅2900Truncated Birth-Death ChainA chain runs on states \ 0,1,\dots,m\ . From interior state i it moves to i+1 with probability p, to i-1 with probability q, and stays put with probability 1-p-q. At 0 the left move is replaced by staying put; at m the right move is replaced by staying put. Assume p,q>0. Find the stationary distribution.概率中等derivation未尝试面试订阅2901A Tiny Queue with Capacity TwoA queue length process lives on \ 0,1,2\ . From state 0 it stays at 0 with probability 1-a and moves to 1 with probability a. From state 1 it moves to 2 with probability a, to 0 with probability b, and stays with probability 1-a-b. From state 2 it moves to 1 with probability b and stays at 2 with probability 1-b. Find the stationary distribution.概率中等derivation未尝试面试订阅2902Laziness Does Not Change StationaritySuppose is stationary for a Markov chain with transition matrix P. Fix \in(0,1) and define a lazy version \[ P'= I+(1- )P. \] Show that is also stationary for P'.概率简单derivation未尝试面试订阅2903Weighted Random Walk with ConductancesOn an undirected graph, each edge \ u,v\ has a positive conductance c uv =c vu . The chain moves from u to v with probability \[ P(u,v)= c uv \sum w c uw . \] Find the stationary distribution.概率中等derivation未尝试面试订阅2904Metropolis Chain for Target Weights 1,2,5On the line graph 1\leftrightarrow 2\leftrightarrow 3, propose moving to a neighboring state uniformly, and accept proposals using the Metropolis rule targeting weights proportional to (1,2,5). What is the stationary distribution?概率中等derivation未尝试面试订阅2905Ehrenfest Urn StationarityIn the Ehrenfest urn model with N balls, state i means exactly i balls are red. Each step, choose one ball uniformly at random and flip its color. Find the stationary distribution of the chain on \ 0,1,\dots,N\ .概率困难derivation未尝试面试订阅2906Uniform Teleportation on Top of a Doubly Stochastic BaseLet Q be a finite doubly stochastic matrix, and let U be the matrix whose every row is uniform over all states. For \[ P= U+(1- )Q,\qquad 0< <1, \] find the stationary distribution of P.概率中等derivation未尝试面试订阅2907Long-Run Occupancy on a 3x3 GridA simple random walk moves on the 3 3 grid graph. What is the stationary probability of the center, of a non-corner edge vertex, and of a corner vertex? What total stationary mass lies on the corners?概率中等derivation未尝试面试订阅