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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2941Two Random Times More Than 20 Seconds ApartTwo independent timestamps are drawn uniformly from the interval [0,60] seconds. What is the probability that they differ by more than 20 seconds?概率中等derivation未尝试面试订阅2942Expected Gap Between Two Random Points on an IntervalIf X and Y are independent Uniform (0,1) points, compute E[|X-Y|].概率中等derivation未尝试面试订阅2943Two Random Points Within 0.1 of Each OtherTwo independent points are drawn uniformly from [0,1]. What is the probability that their distance is less than 0.1?概率简单derivation未尝试面试订阅2944Midpoint Beyond Two ThirdsIf X and Y are independent Uniform (0,1) points, what is the probability that their midpoint exceeds 2/3?概率简单derivation未尝试面试订阅2946Midpoint Inside the Central SquareTwo independent random points are drawn uniformly from the unit square. What is the probability that their midpoint lies in the central square [1/4,3/4] [1/4,3/4]?概率中等derivation未尝试面试订阅2949Expected Distance to the Nearest Vertical SideA random point (X,Y) is uniform on the unit square. Compute the expected distance from the point to the nearer of the two vertical sides.概率简单derivation未尝试面试订阅2950Radial CDF in the Unit DiskA random point is uniform in the unit disk. Let R be its distance from the center. Compute P(R\le r) for 0\le r\le 1.概率简单derivation未尝试面试订阅2951Expected Radius in the Unit DiskA random point is uniform in the unit disk. Let R be its distance from the center. Compute E[R].概率中等derivation未尝试面试订阅2952Expected Squared Radius in the Unit DiskA random point is uniform in the unit disk. Compute E[R 2], where R is its distance from the center.概率简单derivation未尝试面试订阅2953Expected Larger Radius from Two Disk PointsTwo independent random points are drawn uniformly from the unit disk. Let R 1 and R 2 be their distances from the center, and let \[ M=\max(R 1,R 2). \] Compute E[M].概率中等derivation未尝试面试订阅2956Expected x-Coordinate in the Right TriangleA random point is uniform in \[ T=\ (x,y): x\ge 0,\ y\ge 0,\ x+y\le 1\ . \] Compute E[X].概率中等derivation未尝试面试订阅2960Expected Shorter Arc Between Two Circle PointsTwo independent random points are chosen uniformly on the unit circle. Let \Theta be the angular length of the shorter arc between them. Compute E[\Theta].概率中等derivation未尝试面试订阅2963Closer to the Center Than to the Left EdgeA random point (X,Y) is uniform in the unit square. What is the probability that the point is closer to the center (1/2,1/2) than to the left edge x=0?概率困难derivation未尝试面试订阅2964Expected Area of the Rectangle to the OriginA random point (X,Y) is uniform in the unit square. Consider the axis-aligned rectangle with opposite corners at (0,0) and (X,Y). What is the expected area of that rectangle?概率简单derivation未尝试面试订阅2968Expected Shorter Piece from One CutA unit stick is cut once at a uniformly random point. Let S be the length of the shorter piece. Compute E[S].概率简单derivation未尝试面试订阅2969Expected Product of the Two PiecesA unit stick is cut once at a uniformly random point. If the two piece lengths are U and 1-U, compute E[U(1-U)].概率简单derivation未尝试面试订阅