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2816PGF of a Binomial VariableLet X\sim Binomial (n,p). Derive its probability generating function G X(s) and use it to recover E[X].概率中等derivation未尝试面试订阅2817Sum of Independent Poisson CountsLet X\sim Poisson (\lambda 1) and Y\sim Poisson (\lambda 2) be independent. Use PGFs to identify the distribution of X+Y.概率中等derivation未尝试面试订阅2818Poisson ThinningSuppose N\sim Poisson ( ) and each event is independently kept with probability p. Let K be the number kept. Use PGFs to identify the law of K.概率中等derivation未尝试面试订阅2819A Generic Even-Parity FormulaLet X be a nonnegative integer-valued random variable with PGF G X(s). Express P(X is even ) in terms of G X(-1).概率中等derivation未尝试面试订阅2820Even Poisson CountIf N\sim Poisson ( ), use its PGF to compute P(N is even ).概率中等derivation未尝试面试订阅2821Even Binomial CountIf X\sim Binomial (n,p), compute P(X is even ) using the PGF.概率中等derivation未尝试面试订阅2822Extinction for Offspring 0 or 2A Galton-Watson branching process has offspring PGF \phi(s)=0.3+0.7s 2. Compute the extinction probability.概率中等derivation未尝试面试订阅2824Critical 0-or-2 BranchingA branching process has offspring PGF \phi(s)=\frac12+\frac12 s 2. What is the extinction probability?概率中等derivation未尝试面试订阅2825Compound Poisson With Geometric Batch SizeLet N\sim Poisson (2), and conditional on N, let \[ S=\sum i=1 N B i, \] where the B i are i.i.d. geometric-on-\ 1,2,\dots\ with parameter 1/2, so P(B i=k)=2 -k . Find the PGF of S and compute E[S].概率中等derivation未尝试面试订阅2827Generic Thinning of an Arbitrary CountLet X be a nonnegative integer-valued random variable with PGF G X(s). Each of the X items is independently kept with probability p. If Y is the number kept, express G Y(s) in terms of G X.概率中等derivation未尝试面试订阅2828Mean and Variance After ThinningUnder the thinning setup above, derive E[Y] and Var (Y) in terms of E[X] and Var (X).概率中等derivation未尝试面试订阅2829A Geometric Number of Bernoulli TrialsLet N have the geometric law on \ 0,1,2,\dots\ with P(N=n)=p(1-p) n. Conditional on N, let \[ S=\sum i=1 N X i, \] where the X i are i.i.d. Bernoulli(q). Find the PGF of S and identify its distribution.概率中等derivation未尝试面试订阅2830Total Progeny PGF EquationLet \phi(s) be the offspring PGF of a Galton-Watson branching process started from one ancestor, and let T be the total progeny. Show that the PGF of T satisfies \[ G T(s)=s\,\phi(G T(s)). \]概率中等derivation未尝试面试订阅2831Mean Total Progeny in the Subcritical CaseSuppose a Galton-Watson branching process starts from one ancestor and has offspring PGF \phi with mean m=\phi'(1)<1. Use the total-progeny PGF equation to derive E[T].概率中等derivation未尝试面试订阅2832Binomial Number of Trade BatchesLet N\sim Binomial (5,0.4). Conditional on N, let \[ S=\sum i=1 N B i, \] where each batch size B i has PGF H(s)=0.5+0.3s+0.2s 2. Find the PGF of S and compute E[S].概率中等derivation未尝试面试订阅2833Zero Total in a Compound Poisson Batch ModelLet N\sim Poisson (3) and let the i.i.d. batch sizes B i satisfy \[ P(B i=0)=0.2,\quad P(B i=1)=0.5,\quad P(B i=2)=0.3. \] If S=\sum i=1 N B i, compute P(S=0) using the PGF.概率中等derivation未尝试面试订阅2834Two-Stage Thinning Collapses to OneA count variable X is first thinned independently with keep probability p, and then each surviving item is independently kept again with probability q. Show that the final count has PGF G X(1-pq+pq\,s).概率中等derivation未尝试面试订阅2835A Geometric Parent Count With Even-Sized BatchesLet N have the geometric law on \ 0,1,2,\dots\ with P(N=n)=\frac13 (\frac23 ) n. Conditional on N, let \[ S=\sum i=1 N B i, \] where P(B i=0)=P(B i=2)=1/2. Compute P(S=4) using the PGF.概率中等derivation未尝试面试订阅2836Deterministic Batch Size TwoLet N\sim Poisson ( ) and define S=2N. What is the PGF of S? What are P(S odd ) and E[S]?概率中等derivation未尝试面试订阅2837Immigration Plus Branching in One StepLet Z t be a branching process with immigration. Each individual in generation t produces offspring with PGF \phi(s), independently, and the number of immigrants arriving at the next generation has PGF \psi(s), independently of everything else. Express the PGF of Z t+1 in terms of the PGF of Z t.概率中等derivation未尝试面试订阅