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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2929Survival from Five FoundersA branching process has extinction probability q=3/7 from one ancestor. If the process starts from five 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2930Expected Fourth Generation under Mild SupercriticalityA branching process starts from Z 0=2 ancestors and has mean offspring count m=1.3. What is E[Z 4]?概率简单derivation未尝试面试订阅2931Expected Fifth Generation in a Subcritical ProcessA branching process starts from 10 ancestors and has mean offspring count m=0.8. What is E[Z 5]?概率简单derivation未尝试面试订阅2932Conditional Variance One Step AheadIn a branching process, let the offspring distribution have variance 2. Express \[ Var (Z n+1 \mid Z n) \] in terms of Z n.概率简单derivation未尝试面试订阅2933Variance Recursion across GenerationsLet a branching process have offspring mean m and variance 2. Show that \[ Var (Z n+1 )=m 2 Var (Z n)+ 2 E[Z n]. \]概率中等derivation未尝试面试订阅2934Expected Population Stays Flat in the Critical CaseA branching process starts from one ancestor and has mean offspring count m=1. What is E[Z n]?概率简单derivation未尝试面试订阅2935Positive Mean Growth Does Not Guarantee Survival on a Given PathA branching process has mean offspring count 1.1 but extinction probability q=2/3. Explain, in one or two sentences, why there is no contradiction between these two facts.概率中等derivation未尝试面试订阅2936Extinction Probability from k FoundersIf the extinction probability from one ancestor is q, what is the extinction probability from k independent ancestors?概率简单derivation未尝试面试订阅