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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4216Normalized MDI Share 1A random forest reports total mean-decrease-in-impurity contributions spread=0.42, imbalance=0.21, id bucket=0.07. What are the normalized importance shares, and which feature ranks first?机器学习简单数值题未尝试面试订阅4217Normalized MDI Share 2A model has baseline validation AUC 0.62. After permuting three features separately, AUC becomes 0.57 for value signal, 0.60 for momentum, and 0.61 for zip code. What permutation-importance drops do these imply, and which feature ranks first?机器学习简单数值题未尝试面试订阅4218Normalized MDI Share 3A sector feature is represented by three one-hot columns with impurity-gain importances 0.04, 0.03, and 0.01. Two other features have importances 0.05 and 0.07. If you aggregate the one-hot block into a single group, what are the normalized group shares and which group ranks first?机器学习简单数值题未尝试面试订阅4219Normalized MDI Share 4Two trees contribute split gains to features A and B. Tree 1 contributes A=12, B=5. Tree 2 contributes A=8, B=10. What are the total normalized gain importances for A and B?机器学习简单数值题未尝试面试订阅4220Normalized MDI Share 5A model has baseline log loss 0.400. After permuting feature X, log loss rises to 0.455; after permuting feature Y, it rises to 0.420. What are the permutation importances under a log-loss metric, and which feature is more important?机器学习简单数值题未尝试面试订阅4221Grouped Permutation Drop Pattern 1A model starts at validation accuracy 0.82. Permuting feature X1 alone lowers it to 0.79, permuting X2 alone lowers it to 0.8, and permuting them together lowers it to 0.7. What are the three drops, and what pattern does that suggest?机器学习中等数值题未尝试面试订阅4223Grouped Permutation Drop Pattern 3A model has baseline AUC 0.70. With a correlated twin present, permuting feature A drops AUC to 0.64. After removing the twin, permuting A drops AUC to 0.58. By how much did feature A's permutation importance increase?机器学习中等数值题未尝试面试订阅4224Grouped Permutation Drop Pattern 4An impurity-based feature ranking is id hash=0.40, signal 1=0.35, signal 2=0.25. After limiting max depth, id hash gain is cut in half while the other raw gains are unchanged. What are the new normalized shares?机器学习中等数值题未尝试面试订阅