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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2908How Often Is a Given Edge Crossed?A simple random walk on a connected undirected graph is started in stationarity. What is the long-run probability that the next step traverses a particular undirected edge \ u,v\ , regardless of direction?概率困难derivation未尝试面试订阅2910Simple Walk on a Three-State PathConsider the path graph 1\leftrightarrow 2\leftrightarrow 3. A simple random walk moves to a uniformly chosen neighbor. Find the stationary distribution.概率简单derivation未尝试面试订阅2913Half the Mass on Each Side of K_{m,n}A simple random walk runs on the complete bipartite graph K m,n . Show that although the two sides have different numbers of vertices, each side carries stationary mass 1/2.概率中等derivation未尝试面试订阅2914A Concrete Weighted TriangleA weighted random walk runs on vertices \ 1,2,3\ with symmetric conductances \[ c 12 =1,\qquad c 23 =2,\qquad c 13 =3. \] From a vertex, the chain picks the next vertex proportionally to the conductance of the connecting edge. Find the stationary distribution.概率中等derivation未尝试面试订阅2915Random Walk on the HypercubeConsider the chain on \ 0,1\ d that chooses one coordinate uniformly at random at each step and flips that bit. Find the stationary distribution.概率中等derivation未尝试面试订阅2916Mean Generation Size from One FounderA Galton-Watson process starts from one ancestor, and each individual has offspring mean m. What is E[Z n]?概率简单derivation未尝试面试订阅2917Mean Generation Size from k FoundersA branching process starts from Z 0=k ancestors and has offspring mean m. What is E[Z n]?概率简单derivation未尝试面试订阅2918One-Step Conditional ExpectationIn a Galton-Watson process with offspring mean m, express E[Z n+1 \mid Z n] in terms of Z n.概率简单derivation未尝试面试订阅2919The Standard Branching MartingaleLet m>0 be the offspring mean in a Galton-Watson process. Show that \[ M n= Z n m n \] is a martingale with respect to the natural filtration.概率中等derivation未尝试面试订阅2920Expected Total Progeny in the Subcritical CaseA Galton-Watson process starts from one ancestor and has offspring mean m<1. Let \[ T=\sum n\ge 0 Z n \] be the total progeny. Compute E[T].概率中等derivation未尝试面试订阅2921Expected Total Progeny from k FoundersA subcritical branching process with offspring mean m<1 starts from k ancestors. What is the expected total progeny?概率简单derivation未尝试面试订阅2922Exact Extinction Probability for 0-or-2 OffspringEach individual has 0 children with probability 0.3 and 2 children with probability 0.7. Starting from one ancestor, compute the extinction probability.概率中等derivation未尝试面试订阅2923Another Exact Extinction RootEach individual has offspring distribution \[ P(\xi=0)=0.2,\qquad P(\xi=1)=0.5,\qquad P(\xi=2)=0.3. \] Starting from one ancestor, compute the extinction probability.概率中等derivation未尝试面试订阅2924Critical 0-or-2 BranchingSuppose each individual has 0 children with probability 1/2 and 2 children with probability 1/2. What is the extinction probability?概率中等derivation未尝试面试订阅