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2979Leftmost Piece Exceeds One HalfTwo cut points are drawn independently and uniformly on [0,1] and ordered. Let L be the leftmost piece length. What is P(L>1/2)?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2981Size-Biased Piece from Two CutsTwo cut points are drawn independently and uniformly on [0,1], forming three pieces with lengths X 1,X 2,X 3. A random point is then drawn uniformly on the stick, and you keep the piece containing that point. Let B be its length. Compute E[B].概率困难derivation未尝试面试订阅2982Second Moment of a Size-Biased PieceIn the same setup as above, a random point is drawn on the stick after two cuts and B is the length of the piece containing that point. Compute E[B 2].概率困难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2984Longest Piece Below One Half with Three CutsThree cut points are drawn independently and uniformly on [0,1] and ordered, producing four pieces. What is the probability that the longest piece has length less than 1/2?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2986Expected Longest Piece with Three CutsThree cut points are drawn independently and uniformly on [0,1] and ordered, producing four pieces. Let L be the longest piece length. Compute E[L].概率困难derivation未尝试面试订阅2987Expected Sum of Squared Lengths with Three CutsThree cut points are drawn independently and uniformly on [0,1] and ordered, producing four pieces with lengths X 1,\dots,X 4. Compute E[X 1 2+\cdots+X 4 2].概率中等derivation未尝试面试订阅2988Expected Length of a Size-Biased Piece with Three CutsThree cut points are drawn independently and uniformly on [0,1]. After ordering them, a random point is chosen uniformly on the stick and you keep the piece containing that point. Let B be that piece length. Compute E[B].概率困难derivation未尝试面试订阅2989Expected Shortest Piece with Five PiecesFour cut points are drawn independently and uniformly on [0,1] and ordered, producing five pieces. Let M be the shortest piece length. Compute E[M].概率困难derivation未尝试面试订阅2990Expected Longest Piece with Five PiecesFour cut points are drawn independently and uniformly on [0,1] and ordered, producing five pieces. Let L be the longest piece length. Compute E[L].概率困难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3001Exactly One in the First Half Hour and None in the SecondA Poisson process has rate 8 per hour. What is the probability of exactly one arrival in the first 30 minutes and zero arrivals in the next 30 minutes?概率简单derivation未尝试面试订阅3002Two in the First Ten Minutes and Zero in the Next FiveA Poisson process has rate 24 per hour. What is the probability of exactly two arrivals in the first 10 minutes and zero arrivals in the next 5 minutes?概率简单derivation未尝试面试订阅3003No Messages Early but At Least One LaterA Poisson process has rate 15 per hour. What is the probability of zero arrivals in the first 4 minutes and at least one arrival in the next 8 minutes?概率简单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3016Buy Orders Beat Sell OrdersBuy orders and sell orders arrive as independent Poisson processes with rates \lambda A=7 and \lambda B=5 per hour. What is the probability that the next arrival in the merged stream comes from stream A?概率简单derivation未尝试面试订阅