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2752Winner's Curse After Seeing High Against One Low SignalAn asset has common value V\in\ 0,100\ with prior P(V=100)=1/2. Each bidder receives a signal that equals the true state with probability 0.8, independently across bidders conditional on V. You observe a high signal and learn that your rival observed a low signal. What is E[V\mid your signal high, rival low ]? If you bid 80 in a first-price auction and win for sure in this information set, what is your expected profit?脑筋急转弯中等derivation未尝试面试订阅2753Winner's Curse Gets Stronger Against Two Low SignalsUse the same binary common-value setup as above, except now there are three bidders total. You observe a high signal and learn that the other two bidders both observed low signals. Compute E[V\mid your signal high, both rivals low ].脑筋急转弯中等derivation未尝试面试订阅2765Why Private Values Avoid the Winner's CurseExplain why the winner's curse is a feature of common-value auctions but not of independent private-value auctions. Your answer should make clear what information winning reveals in each environment.脑筋急转弯简单derivation未尝试面试订阅2803Conditioning on the Outer Half of the DiskA point is chosen uniformly in the unit disk. Conditional on the point lying outside the disk of radius 1/2, what is its expected distance from the center?脑筋急转弯中等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2857A Two-Volatility Mixture Is Not GaussianA return R is conditionally Gaussian: \[ R\mid V= \sim N(0, 2), \] where V equals 1 or 2 with probability 1/2 each. Compute the characteristic function of R and explain why R is not itself Gaussian.概率中等derivation未尝试面试订阅2863A Batch-Size Compound Poisson Desk FlowTrades arrive according to N\sim Poisson ( ). Each trade contributes a batch size B taking values 0,1,2 with probabilities 1/2,1/3,1/6, independently across trades and from N. Let \[ S=\sum k=1 N B k. \] Find the MGF of S, and compute E[S] and Var (S).概率中等derivation未尝试面试订阅2864Exponential Random Intensity Gives Geometric CountsA latent intensity \Lambda is Exponential ( ) with rate . Conditional on \Lambda, the count N is Poisson (\Lambda). Use MGFs to identify the unconditional law of N, and compute E[N].概率中等derivation未尝试面试订阅2916Mean Generation Size from One FounderA Galton-Watson process starts from one ancestor, and each individual has offspring mean m. What is E[Z n]?概率简单derivation未尝试面试订阅2917Mean Generation Size from k FoundersA branching process starts from Z 0=k ancestors and has offspring mean m. What is E[Z n]?概率简单derivation未尝试面试订阅