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2970Expected Absolute Imbalance After One CutA unit stick is cut once at a uniformly random point. Let D be the absolute difference between the two resulting lengths. Compute E[D].概率简单derivation未尝试面试订阅2971Three Pieces All Longer Than One FifthTwo cut points are drawn independently and uniformly on [0,1], and the stick is cut at the ordered locations. What is the probability that all three resulting pieces have length greater than 1 5 ?概率中等derivation未尝试面试订阅2973Triangle Condition for a Broken StickTwo cut points are drawn independently and uniformly on [0,1], and the stick is cut into three pieces. What is the probability that the three pieces can form a triangle?概率中等derivation未尝试面试订阅2975Expected Shortest Piece with Two CutsTwo cut points are drawn independently and uniformly on [0,1], then ordered to form three stick pieces with lengths X 1,X 2,X 3. Compute E[\min(X 1,X 2,X 3)].概率中等derivation未尝试面试订阅2980Expected Leftmost Piece LengthTwo cut points are drawn independently and uniformly on [0,1] and ordered. Let L be the leftmost piece length. Compute E[L].概率中等derivation未尝试面试订阅2983Four Pieces All Longer Than One SixthThree cut points are drawn independently and uniformly on [0,1] and ordered, producing four pieces. What is the probability that all four pieces are longer than 1/6?概率中等derivation未尝试面试订阅2985Expected Shortest Piece with Three CutsThree cut points are drawn independently and uniformly on [0,1] and ordered, producing four pieces. Let M be the shortest piece length. Compute E[M].概率中等derivation未尝试面试订阅2991Quiet Ten-Minute Window RateEvents follow a Poisson process with hourly rate . What value of makes the probability of zero events in the next 10 minutes exactly 0.7?概率简单derivation未尝试面试订阅2996First Arrival Before Market Open Plus Two MinutesMessages arrive as a Poisson process with rate 9 per hour. What is the probability the first arrival occurs within 2 minutes?概率中等derivation未尝试面试订阅3000First Order Before Twenty SecondsMessages arrive as a Poisson process with rate 45 per hour. What is the probability the first arrival occurs within 0.3333333333333333 minutes?概率中等derivation未尝试面试订阅3011Median Waiting Time for a One-Per-Hour FeedArrivals follow a Poisson process with rate 1 per hour. What is the median waiting time to the first arrival, in minutes?概率简单derivation未尝试面试订阅3261Area of an Ellipse from the Unit DiskThe map x=2u,\ y=3v sends the unit disk u 2+v 2\le 1 to an ellipse in the (x,y)-plane. What is the area of that ellipse?数学中等derivation未尝试面试订阅3262Area of a Parallelogram from a Unit SquareThe map x=u+v,\ y=u-v sends the unit square 0\le u,v\le 1 to a parallelogram. What is its area?数学中等derivation未尝试面试订阅3263Integral After a Simple ScalingUse the change of variables x=2u,\ y=3v to compute \iint R 1\,dx\,dy, where R is the image of the rectangle 0\le u\le 1,\ 0\le v\le 2.数学中等derivation未尝试面试订阅3264Area Scaling Under a Triangular MapThe map x=u,\ y=u+2v sends the unit square to a parallelogram. Compute the image area.数学中等derivation未尝试面试订阅3265Area of a Polar WedgeUse the polar Jacobian to compute the area of the region 0\le r\le 2,\ 0\le \le /3.数学中等derivation未尝试面试订阅3368Midpoint Rule Exact on a LineUsing a single-panel midpoint rule, approximate \int 0 4 (1+x)\,dx.数学简单derivation未尝试面试订阅3369Trapezoid Rule on a Symmetric Convex FunctionUsing a single-panel trapezoid rule, approximate \int -1 1 x 2\,dx.数学简单derivation未尝试面试订阅3370Midpoint Rule Misses a Symmetric BowlUsing a single-panel midpoint rule, approximate \int -1 1 x 2\,dx.数学简单derivation未尝试面试订阅3371Simpson on a CubicUsing one Simpson panel, approximate \int 0 2 x 3\,dx.数学中等derivation未尝试面试订阅