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325Robust Comparable Pairs in Random PointsLet X 1, X 2, \dots, X n be independent and uniformly distributed on [0,1] d (the d-dimensional unit hypercube). Two points X i and X j are called comparable if one dominates the other coordinatewise, i.e., either X i \le X j in every coordinate or X j \le X i in every coordinate. Find the expected number of comparable pairs. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费335Robust Variance of the Sample Mean Under Sampling Without ReplacementAn urn contains N balls numbered 1, 2, \dots, N. You draw n balls without replacement and let X = \tfrac 1 n \sum i=1 n X i, where X i is the number on the i-th draw. Derive Var ( X ) in terms of N and n, and evaluate it for N = 10, n = 4. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率困难derivation未尝试免费343Robust Covariance of Multinomial CountsA fair six-sided die is rolled 60 times independently. Let N 1 be the number of times face 1 appears and N 2 the number of times face 2 appears. (a) Find Cov (N 1, N 2). (b) Use your answer to compute Var (N 1 + N 2) and verify it by recognizing the distribution of N 1 + N 2. Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.概率中等数值题未尝试免费372Expected Maximum of Correlated Bernoullis via Indicator and TowerLet U \sim Uniform (0,1) and, given U, let X and Y be conditionally i.i.d.\ Bernoulli (U). Define M = \max(X, Y). Using the tower property and the indicator representation M = 1 \ X \ge 1 or Y \ge 1\ , find E[M].概率简单数值题未尝试免费405Joint Distribution of Extremes and the RangeLet X 1, \ldots, X n be iid Uniform (0,1). Let X (1) = \min i X i and X (n) = \max i X i.概率困难multi part未尝试面试订阅413Beta Distribution of the k-th Uniform Order StatisticLet X 1, \ldots, X n be iid Uniform (0,1). Derive that the k-th order statistic X (k) has the Beta (k, n-k+1) distribution.概率中等derivation未尝试免费414Renyi Representation of Exponential Order-Statistic SpacingsLet X 1, \ldots, X n be iid Exp ( ) and let X (1) \le \cdots \le X (n) be the order statistics. Define the normalized spacings D k = (n-k+1)(X (k) - X (k-1) ) for k = 1, \ldots, n, where X (0) = 0.概率困难multi part未尝试面试订阅419Conditional Distribution of the Minimum Given the MaximumLet X 1, \ldots, X n be iid Uniform (0,1) with n \ge 3. Let X (1) and X (n) denote the minimum and maximum.概率困难multi part未尝试面试订阅425Ratio of the Two Smallest Exponential Order StatisticsLet X 1, X 2 be independent Exp (1) random variables with order statistics X (1) \le X (2) . Define U = X (1) / X (2) .概率困难multi part未尝试面试订阅566Time to Reach 3 Distinct Types 1Coupons arrive uniformly from 7 types. What is the expected number of draws needed to see 3 distinct types for the first time?概率简单数值题未尝试免费2654Expected Number of Times a Point Is Validated in Repeated k-Fold CVIn R repeats of ordinary k-fold CV, each point appears in exactly one validation fold per repeat. Derive the number of validation appearances of one point across all repeats.机器学习中等derivation未尝试面试订阅2660Why Rare-Event Stratification MattersWhy can ordinary random folds become misleading in a rare-event problem even when the data are IID?机器学习困难essay未尝试面试订阅2664How Many Distinct Hyperparameter Winners Can Outer CV ProduceA nested CV uses 7 outer folds and selects exactly one hyperparameter setting inside each outer fold. What is the maximum possible number of distinct winning hyperparameter settings across outer folds?机器学习困难derivation未尝试面试订阅2701Search Depth Hidden in Design KnobsA team tried 4 universes, 5 rebalance frequencies, and 6 transaction-cost assumptions before reporting the best result. How many distinct design combinations were implicitly searched?机器学习简单数值题未尝试面试订阅2714Why Search Depth Is Bigger Than the Number of Named StrategiesWhy can a team that claims to have tested only five named strategies still have conducted a much deeper search than that number suggests?机器学习困难essay未尝试面试订阅5946Halfway CollectionTrading cards come in 10 equally likely types, independent across packs. What is the expected number of packs needed until you own 5 distinct types (any 5, not a specific set)?概率中等数值题未尝试免费5948Empty Mailboxes10 letters are placed independently and uniformly at random into 8 mailboxes. What is the expected number of mailboxes that remain empty?概率简单数值题未尝试免费5950Variance of the Coverage Count4 balls are thrown independently and uniformly into 6 boxes. Let D be the number of boxes that receive at least one ball. Compute Var (D).概率困难数值题未尝试面试订阅5952Distinct Types Across Two PacksA booster pack contains 4 cards drawn WITHOUT replacement from a pool of 9 equally likely distinct types (so the 4 cards in a single pack are all different types). You open two packs; the two packs are independent of each other (8 cards total). What is the expected number of DISTINCT types you own?概率简单数值题未尝试免费5956How Much by the DeadlineYou will draw exactly 6 coupons, each uniform and independent over 4 types. The promotion ends after these 6 draws. What is the expected number of DISTINCT types you will have collected by the deadline?概率简单数值题未尝试免费