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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2838Sum of Geometric CountsLet X 1,\dots,X r be i.i.d. geometric random variables on \ 0,1,2,\dots\ with \[ P(X i=k)=p(1-p) k. \] Use PGFs to identify the distribution of S=X 1+\cdots+X r and compute E[S].概率中等derivation未尝试面试订阅2840Poisson Number of Geometric Ticket GeneratorsLet N\sim Poisson (3). Conditional on N, let \[ S=\sum i=1 N B i, \] where each B i is geometric on \ 0,1,2,\dots\ with success parameter p=1/4, so P(B i=k)=\frac14(\frac34) k. Find the PGF of S and compute E[S].概率中等derivation未尝试面试订阅2841Signed Compound Poisson Order FlowMarket buys arrive as +1 and sells as -1. The number of trades in a minute is N\sim Poisson ( ), and conditional on a trade, the sign is +1 with probability p and -1 with probability 1-p, independently. Let S be the net signed flow in that minute. Find the MGF of S, and compute E[S] and Var (S).概率中等derivation未尝试面试订阅2842Compound Poisson with Exponential SeveritiesClaims arrive according to N\sim Poisson ( ). Claim sizes X 1,X 2,\dots are i.i.d. Exponential ( ) with rate , independent of N. Let S=\sum i=1 N X i. Derive the MGF of S, and compute E[S] and Var (S).概率中等derivation未尝试面试订阅2843Gamma-Poisson Mixing Produces Negative Binomial CountsA latent intensity \Lambda is Gamma ( , ) with shape and rate . Conditional on \Lambda, the count N is Poisson (\Lambda). Use MGFs to identify the unconditional distribution of N.概率困难derivation未尝试面试订阅2844A Geometric Number of Exponential StagesA task takes a geometric number of stages: N has support \ 1,2,\dots\ with P(N=n)=p(1-p) n-1 . Each stage duration is i.i.d. Exponential ( ), independent of N. Let T=\sum i=1 N X i. Use MGFs to identify the law of T.概率困难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2856Compound Poisson with Gaussian JumpsLet N\sim Poisson ( ) and let Y 1,Y 2,\dots be i.i.d. N( , 2), independent of N. For \[ S=\sum k=1 N Y k, \] derive the MGF of S, and compute E[S] and Var (S).概率中等derivation未尝试面试订阅