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651Fair Corridor Hit Probability 1A symmetric random walk starts at 2 on the integer line and stops when it first hits 0 or 7. What is the probability that it hits 7 before 0?概率简单derivation未尝试免费656Fair Corridor Exit Time 1A symmetric random walk starts at 1 and stops when it first hits 0 or 6. What is the expected stopping time?概率中等数值题未尝试免费661Lazy Walk Exit Time 1A lazy symmetric walk starts at 2 and, each period, moves +1 with probability 3/8, moves -1 with probability 3/8, and stays put with probability 1/4. It stops when it first hits 0 or 8. What is the expected stopping time?概率简单数值题未尝试免费667Biased Corridor Hit Probability 1A random walk starts at 2, moves +1 with probability 3/5 and -1 with probability 2/5, and stops when it first hits 0 or 7. What is the probability that it reaches 7 before 0?概率中等数值题未尝试免费673Scaled-Step Exit Time 3A fair random walk starts at 3 and moves by +3 or -3 with equal probability each step. It stops when it first hits -9 or 9. What is the expected stopping time?概率中等数值题未尝试免费674Scaled-Step Exit Time 2A fair random walk starts at 4 and moves by +4 or -4 with equal probability each step. It stops when it first hits 0 or 16. What is the expected stopping time?概率中等derivation未尝试免费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