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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2938A Supercritical Cubic with a Clean RootEach individual has 0 children with probability 1/2 and 3 children with probability 1/2. Starting from one ancestor, compute the extinction probability.概率困难derivation未尝试面试订阅2939Another Supercritical Two-Point LawEach individual has 0 children with probability 0.4 and 2 children with probability 0.6. Compute the extinction probability.概率中等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2941Two Random Times More Than 20 Seconds ApartTwo independent timestamps are drawn uniformly from the interval [0,60] seconds. What is the probability that they differ by more than 20 seconds?概率中等derivation未尝试面试订阅2943Two Random Points Within 0.1 of Each OtherTwo independent points are drawn uniformly from [0,1]. What is the probability that their distance is less than 0.1?概率简单derivation未尝试面试订阅2944Midpoint Beyond Two ThirdsIf X and Y are independent Uniform (0,1) points, what is the probability that their midpoint exceeds 2/3?概率简单derivation未尝试面试订阅2945Coordinatewise Comparability in the Unit SquareTwo independent random points (X 1,Y 1) and (X 2,Y 2) are drawn uniformly from the unit square. What is the probability that one point is coordinatewise less than the other, i.e. either X 1<X 2 and Y 1<Y 2, or X 1>X 2 and Y 1>Y 2?概率中等derivation未尝试面试订阅2946Midpoint Inside the Central SquareTwo independent random points are drawn uniformly from the unit square. What is the probability that their midpoint lies in the central square [1/4,3/4] [1/4,3/4]?概率中等derivation未尝试面试订阅2948Closer to Vertical Sides Than Horizontal SidesA random point is drawn uniformly from the unit square. What is the probability that its distance to the nearer vertical side is smaller than its distance to the nearer horizontal side?概率中等derivation未尝试面试订阅