第 1 / 2 页
非代码面试题
显示 20 / 22 道匹配题目
答题状态：未尝试未正确已正确
ID题目领域难度题型进度权限
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2858Reading a Laplace Law from Its Characteristic FunctionSuppose a centered random variable has characteristic function \[ \phi X(u)= 1 1+b 2u 2 . \] Identify the law of X, and determine its MGF on the domain where it exists.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2860Characteristic Function of a Uniform Return ShockLet X\sim Uniform [-a,a]. Compute its characteristic function and recover Var (X) from the transform.概率中等derivation未尝试面试订阅2861Why the Difference of Two Copies Is Automatically SymmetricLet X and Y be i.i.d. with characteristic function \phi(u). Show that the characteristic function of D=X-Y is |\phi(u)| 2, and conclude that D is symmetric about 0.概率中等derivation未尝试面试订阅2862Factorized Joint MGF Means IndependenceSuppose \[ M X,Y (s,t)=\exp\! (s+2t+ s 2 2 +2t 2 ). \] Identify the marginal laws of X and Y, and determine whether they are independent.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