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2795Four Points in a Common SemicircleFour points are chosen independently and uniformly on the unit circle. What is the probability that all four lie inside some semicircle?脑筋急转弯中等derivation未尝试面试订阅2797Expected Squared Distance in the Unit DiskTwo points are chosen independently and uniformly from the unit disk. What is the expected squared Euclidean distance between them?脑筋急转弯中等derivation未尝试面试订阅2798Average Manhattan Distance in the Unit SquareTwo points are chosen independently and uniformly in the unit square. What is their expected Manhattan distance |X 1-X 2|+|Y 1-Y 2|?脑筋急转弯中等derivation未尝试面试订阅2799A Unit-Interval Distance WindowIf U and V are independent Uniform (0,1) variables, what is the probability that |U-V|\le 1/4? Solve it geometrically.脑筋急转弯中等derivation未尝试面试订阅2800Average Distance to a Side of an Equilateral TriangleA point is chosen uniformly inside an equilateral triangle of side length 2. What is its expected distance to a fixed side?脑筋急转弯中等derivation未尝试面试订阅2801Staying Away From All Three SidesA point is chosen uniformly inside an equilateral triangle. What is the probability that its distance to each of the three sides is at least one-sixth of the altitude?脑筋急转弯中等derivation未尝试面试订阅2802Closer to the Diagonal Than to the Left EdgeA point is chosen uniformly in the unit square. What is the probability that it is closer to the diagonal line y=x than to the left edge x=0?脑筋急转弯困难derivation未尝试面试订阅2804Chord Longer Than √2Fix one point on the unit circle and choose a second point uniformly on the circle. What is the probability that the chord length exceeds \sqrt2?脑筋急转弯中等derivation未尝试面试订阅2806Closer to the Center Than to the BoundaryA point is chosen uniformly in the unit disk. What is the probability that the point is closer to the center than to the boundary of the disk?脑筋急转弯中等derivation未尝试面试订阅2808Variance of a Coordinate in the DiskA point is chosen uniformly from the unit disk. Let X be its x-coordinate. Compute Var (X).脑筋急转弯中等derivation未尝试面试订阅2809Maximum Coordinate of Two Uniform NumbersTwo independent Uniform (0,1) variables U and V are drawn. What is E[\max(U,V)]? Give a geometric interpretation in the unit square.脑筋急转弯中等derivation未尝试面试订阅2810Expected Squared Chord LengthTwo points are chosen independently and uniformly on the unit circle. What is the expected squared length of the chord joining them?脑筋急转弯中等derivation未尝试面试订阅2811Expected Chord Length From a Uniform MidpointA chord of the unit circle is generated by choosing its midpoint uniformly in the disk. What is the expected chord length?脑筋急转弯中等derivation未尝试面试订阅2813Largest Barycentric Piece Exceeds HalfA point is chosen uniformly inside a triangle. Joining the point to the three vertices partitions the triangle into three smaller triangles. What is the probability that the largest of those three area fractions exceeds 1/2?脑筋急转弯中等derivation未尝试面试订阅2814Closer to the Center Than to the OriginA point is chosen uniformly in the unit square. What is the probability that it is closer to the center (1/2,1/2) than to the origin (0,0)?脑筋急转弯中等derivation未尝试面试订阅2891Two-State Regime SwitchingA market regime is either Calm or Volatile. From Calm, the chain moves to Volatile with probability ; from Volatile, it moves to Calm with probability . Find the stationary distribution.概率简单derivation未尝试面试订阅2892A Three-State Birth-Death Regime ChainConsider states Bull, Neutral, and Bear with transition matrix \[ P= \begin pmatrix 0.8 & 0.2 & 0\\ 0.3 & 0.4 & 0.3\\ 0 & 0.2 & 0.8 \end pmatrix . \] Find the stationary distribution.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