第 1 / 1 页
非代码面试题
显示 11 / 11 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
220Poisson Limit of the Binomial via Characteristic FunctionsLet X n \sim Binomial (n, /n) for fixed > 0 and n > . (a) Write down the characteristic function \varphi X n (t) = E[e itX n ] in closed form. (b) Show that \lim n \varphi X n (t) = e (e it - 1) for every t \in R . (c) Identify the limiting characteristic function and state the convergence-in-distribution conclusion. (d) Justify why pointwise convergence of characteristic functions implies convergence in distribution (cite the relevant theorem). (e) For = 5, n = 100: compute P(X n = 3) using both the exact Binomial PMF and the Poisson approximation, and find the relative error.概率困难derivation未尝试免费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2865A Stable Law with No MGFSuppose X 1,X 2,\dots are i.i.d. with characteristic function \[ \phi X(u)=\exp(-c|u| 3/2 ),\qquad c>0. \] Show that \[ n -2/3 (X 1+\cdots+X n) \] has the same distribution as X 1.概率困难derivation未尝试面试订阅