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2810Expected Squared Chord LengthTwo points are chosen independently and uniformly on the unit circle. What is the expected squared length of the chord joining them?脑筋急转弯中等derivation未尝试面试订阅2811Expected Chord Length From a Uniform MidpointA chord of the unit circle is generated by choosing its midpoint uniformly in the disk. What is the expected chord length?脑筋急转弯中等derivation未尝试面试订阅2812Expected Squared Distance in the Unit SquareTwo points are chosen independently and uniformly in the unit square. What is the expected squared Euclidean distance between them?脑筋急转弯中等derivation未尝试面试订阅2813Largest Barycentric Piece Exceeds HalfA point is chosen uniformly inside a triangle. Joining the point to the three vertices partitions the triangle into three smaller triangles. What is the probability that the largest of those three area fractions exceeds 1/2?脑筋急转弯中等derivation未尝试面试订阅2814Closer to the Center Than to the OriginA point is chosen uniformly in the unit square. What is the probability that it is closer to the center (1/2,1/2) than to the origin (0,0)?脑筋急转弯中等derivation未尝试面试订阅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