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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2909A Four-State Ring with Resets to ZeroConsider states \ 0,1,2,3\ . From state i, the chain moves to i+1 \pmod 4 with probability 1/2 and resets to 0 with probability 1/2. Find the stationary distribution.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2911Custom Three-State Reversible ChainConsider a chain on states \ 1,2,3\ with transition matrix \[ P=\begin pmatrix 0.6 & 0.4 & 0\\ 0.2 & 0.5 & 0.3\\ 0 & 0.6 & 0.4 \end pmatrix . \] Find the stationary distribution.概率中等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2925No Chance of Extinction without Zero OffspringSuppose every individual has at least one child almost surely. What is the extinction probability of the branching process started from one ancestor?概率简单derivation未尝试面试订阅2926Bernoulli Offspring Is Almost Surely Extinct Unless DeterministicSuppose each individual has one child with probability p and zero children with probability 1-p. What is the extinction probability?概率简单derivation未尝试面试订阅2927Poisson Offspring: When Is Survival Possible?If the offspring distribution is Poisson ( ), write the extinction equation and state when the survival probability is positive.概率中等derivation未尝试面试订阅2928Extinction from Three Independent FoundersA branching process started from one ancestor has extinction probability q=2/3. If it starts instead from three independent ancestors, what is the probability that at least one lineage survives forever?概率简单derivation未尝试面试订阅