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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2866Markov Bound for Daily SlippageA nonnegative slippage random variable L has mean E[L]=2 basis points. Give the best Markov upper bound you can on P(L\ge 10).概率简单derivation未尝试面试订阅2867A Generalized Markov Bound from the Fourth MomentSuppose X is any random variable with E[X 4]=81. Use Markov's inequality on a suitable nonnegative variable to bound P(|X|\ge 6).概率中等derivation未尝试面试订阅2868Recovering Chebyshev from MarkovDerive Chebyshev's inequality from Markov's inequality. In other words, show that for any random variable X with mean and variance 2, \[ P(|X- |\ge a)\le 2 a 2 . \]概率简单derivation未尝试面试订阅2870Which Bound Is Better Here?A nonnegative random variable X satisfies E[X]=1 and Var (X)=4. Compare the Markov and Chebyshev upper bounds on P(X\ge 5), and say which one is tighter.概率中等derivation未尝试面试订阅2872How Many Bernoulli Trials for 99% Accuracy?You estimate a Bernoulli success probability by the sample mean of n i.i.d. trials. How large must n be to guarantee, via Hoeffding's inequality, that \[ P(| X n-p|\ge 0.02)\le 0.01? \]概率中等derivation未尝试面试订阅2879Numerical Poisson Overload BoundAn exchange gateway receives N\sim Poisson (100) messages in a fixed interval. Use the Poisson Chernoff upper-tail bound to estimate P(N\ge 130).概率中等derivation未尝试面试订阅2881A Numerical Poisson Shortfall BoundIf N\sim Poisson (100), use the lower-tail Chernoff bound to estimate P(N\le 80).概率中等derivation未尝试面试订阅2888Heterogeneous Range Hoeffding BoundIndependent centered shocks satisfy \[ X 1\in[-1,1],\quad X 2\in[-2,2],\quad X 3\in[-3,3],\quad X 4\in[-4,4] \] almost surely. Use Hoeffding's inequality to bound P(X 1+X 2+X 3+X 4\ge 6).概率中等derivation未尝试面试订阅2889How Large Must the Mean Be for a 2x Poisson Spike to Be Rare?For N\sim Poisson ( ), use the upper-tail Chernoff bound to find a sufficient condition on guaranteeing \[ P(N\ge 2 )\le 0.01. \]概率中等derivation未尝试面试订阅2893Uniform Stationarity for a Doubly Stochastic ChainSuppose a finite Markov chain has transition matrix P whose rows and columns both sum to 1. Show that the uniform distribution is stationary.概率简单derivation未尝试面试订阅2894Random Walk on an Undirected GraphA simple random walk moves on a connected undirected graph G=(V,E) by choosing a uniformly random neighbor at each step. Show that the stationary distribution is proportional to degree.概率简单derivation未尝试面试订阅2895Star Graph Long-Run OccupancyA simple random walk runs on a star graph with one hub and m leaves. What fraction of time is spent at the hub in stationarity, and what fraction at each leaf?概率简单derivation未尝试面试订阅2896Complete Bipartite Graph K_{2,3}A simple random walk runs on the complete bipartite graph K 2,3 . What is the stationary probability of each vertex, and what total stationary mass sits on each side of the bipartition?概率中等derivation未尝试面试订阅2897Triangle with a Pendant NodeConsider a simple random walk on the graph with vertices A,B,C,D, where A,B,C form a triangle and D is connected only to A. Find the stationary distribution.概率中等derivation未尝试面试订阅2898Directed Cycle with Clockwise DriftOn the directed cycle with states \ 0,1,\dots,n-1\ , from each state the chain moves clockwise with probability p and counterclockwise with probability 1-p. Show that the stationary distribution is uniform.概率中等derivation未尝试面试订阅2899Reflecting Simple Walk on 0 to mConsider the chain on \ 0,1,\dots,m\ . From an interior state it moves left or right with probability 1/2 each. From 0 it stays put with probability 1/2 and moves to 1 with probability 1/2; from m it stays put with probability 1/2 and moves to m-1 with probability 1/2. Find the stationary distribution.概率中等derivation未尝试面试订阅2900Truncated Birth-Death ChainA chain runs on states \ 0,1,\dots,m\ . From interior state i it moves to i+1 with probability p, to i-1 with probability q, and stays put with probability 1-p-q. At 0 the left move is replaced by staying put; at m the right move is replaced by staying put. Assume p,q>0. Find the stationary distribution.概率中等derivation未尝试面试订阅