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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2845Recognizing a Shifted Poisson from Its MGFA random variable has MGF \[ M X(t)=\exp\!\bigl(2t+3(e t-1)\bigr). \] Identify the law of X, and compute E[X] and Var (X).概率中等derivation未尝试面试订阅2848Reading Covariance from a Joint MGFSuppose \[ M X,Y (s,t)=\exp\!\bigl(2s-t+2s 2+3st+\tfrac52 t 2\bigr). \] Compute E[X], E[Y], Var (X), Var (Y), and Cov (X,Y).概率中等derivation未尝试面试订阅2849Difference of Two ExponentialsLet X,Y\overset i.i.d. \sim Exponential ( ). Use MGFs to identify the law of D=X-Y.概率中等derivation未尝试面试订阅2850Difference of Two Independent Poisson CountsLet X\sim Poisson (\lambda 1) and Y\sim Poisson (\lambda 2) be independent. Find the characteristic function of D=X-Y, and compute E[D] and Var (D).概率中等derivation未尝试面试订阅2851Rademacher CLT through Characteristic FunctionsLet X 1,X 2,\dots be i.i.d. with P(X i=1)=P(X i=-1)=1/2. Show, using characteristic functions, that \[ X 1+\cdots+X n n \Rightarrow N(0,1). \]概率困难derivation未尝试面试订阅2852The Sample Mean of Cauchy VariablesLet X 1,\dots,X n be i.i.d. standard Cauchy variables, whose characteristic function is \phi(u)=e -|u| . Use characteristic functions to show that the sample mean (X 1+\cdots+X n)/n is again standard Cauchy.概率中等derivation未尝试面试订阅2853Poisson to Normal via Centered Characteristic FunctionsLet N \sim Poisson ( ). Show that \[ N - \Rightarrow N(0,1) \quad as \] by working directly with characteristic functions.概率困难derivation未尝试面试订阅2854Rare-Event Binomial to PoissonLet X n\sim Binomial (n, /n) with fixed >0. Use characteristic functions to show that X n\Rightarrow Poisson ( ).概率中等derivation未尝试面试订阅2859MGF of the Sample Mean of ExponentialsLet X 1,\dots,X n be i.i.d. Exponential ( ) with rate , and let \[ X n= 1 n \sum i=1 n X i. \] Find the MGF of X n, and recover E[ X n] and Var ( X n) from it.概率中等derivation未尝试面试订阅