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2978Expected Product of the Three PiecesTwo 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[X 1X 2X 3].概率中等derivation未尝试面试订阅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3051Earliest Arrival After 20 MinutesA Poisson process is observed on [0,1] hours. Conditional on exactly 4 arrivals in that horizon, what is the probability that the earliest arrival occurs after time 1 3 ?概率中等derivation未尝试面试订阅3052Latest Arrival Before 45 MinutesA Poisson process is observed on [0,1] hours. Conditional on exactly 5 arrivals in that horizon, what is the probability that the latest arrival occurs before time 3 4 ?概率中等derivation未尝试面试订阅3053Earliest Arrival After Half the WindowA Poisson process is observed on [0, 3 2 ] hours. Conditional on exactly 3 arrivals in that horizon, what is the probability that the earliest arrival occurs after time 3 4 ?概率中等derivation未尝试面试订阅3231Why a Sharpe Confidence Interval Is Not a Posterior BeliefA PM says, "The frequentist 95% confidence interval for Sharpe is mostly above zero, so there is a 95% chance the true Sharpe is positive." Why is that statement mixing two frameworks?统计中等essay未尝试面试订阅3236Can a p-Value Be Combined with Prior Conviction?A trader says, "I had a strong prior view, and the p-value came out 0.03, so now I can blend the two and call the trade 97% likely." Why is that not a coherent frequentist calculation?统计简单essay未尝试面试订阅3239Large p-Value Does Not Mean High Posterior Null ProbabilityWhy is it dangerous to read a large p-value as strong evidence that the null is true?统计简单essay未尝试面试订阅3280What Does Whitening Mean When the Covariance Is Singular?Suppose \Sigma=egin pmatrix 2&2\2&2\end pmatrix . Explain what can and cannot be whitened, and identify the subspace where a pseudo-whitening transform still makes sense.数学中等essay未尝试面试订阅3286Why Sign Flips of PCs Do Not MatterWhy are principal components only identified up to sign?数学中等essay未尝试面试订阅3288Why Whitening Can Hurt Economic InterpretabilityWhy can whitening be numerically useful while simultaneously making a factor model harder to interpret economically?数学中等essay未尝试面试订阅3336Why High-Frequency Modes Die FirstWhy does the heat equation smooth out small-scale wiggles faster than large-scale structure?数学中等essay未尝试面试订阅3337Maximum Principle IntuitionWhy can the heat equation not spontaneously create a new interior maximum if there is no internal heat source?数学中等essay未尝试面试订阅