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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2937If Each Individual Has One or Two ChildrenSuppose each individual has one child with probability p and two children with probability 1-p. What is the extinction probability?概率简单derivation未尝试面试订阅2940Almost Sure Extinction in Every Subcritical ProcessA Galton-Watson process has offspring mean m<1. What is its extinction probability?概率简单derivation未尝试面试订阅2981Size-Biased Piece from Two CutsTwo cut points are drawn independently and uniformly on [0,1], forming three pieces with lengths X 1,X 2,X 3. A random point is then drawn uniformly on the stick, and you keep the piece containing that point. Let B be its length. Compute E[B].概率困难derivation未尝试面试订阅2982Second Moment of a Size-Biased PieceIn the same setup as above, a random point is drawn on the stick after two cuts and B is the length of the piece containing that point. Compute E[B 2].概率困难derivation未尝试面试订阅2988Expected Length of a Size-Biased Piece with Three CutsThree cut points are drawn independently and uniformly on [0,1]. After ordering them, a random point is chosen uniformly on the stick and you keep the piece containing that point. Let B be that piece length. Compute E[B].概率困难derivation未尝试面试订阅3001Exactly One in the First Half Hour and None in the SecondA Poisson process has rate 8 per hour. What is the probability of exactly one arrival in the first 30 minutes and zero arrivals in the next 30 minutes?概率简单derivation未尝试面试订阅