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2389Why Regime Averaging Can Beat Crude Mixing 19Why can separately estimating conditional means by regime and then averaging them outperform one crude pooled Monte Carlo run?数学困难derivation未尝试面试订阅2390Why Antithetics May Fail on a Highly Nonmonotone Payoff 20Why can antithetic pairing fail to reduce variance for a payoff that oscillates strongly across the state space?数学困难derivation未尝试面试订阅239195% Half-Width From Sample Volatility and Path Count 21A Monte Carlo estimator has sample standard deviation 5 across n=400 paths. Using the normal approximation, what is the 95% half-width?数学简单数值题未尝试免费2393RMSE of an Unbiased Monte Carlo Estimate 23An unbiased estimator has standard deviation 4 over one path and uses n=64 independent paths. What is the RMSE of the sample mean?数学中等数值题未尝试免费2394Confidence Interval From Mean and Standard Error 24A Monte Carlo estimate is 12.0 with standard error 0.4. Using the normal approximation, what 95% confidence interval do you report?数学中等数值题未尝试免费2409Why More Data Usually Helps a Variance-Dominated Model FirstWhy does collecting more data usually help a high-variance model more than a high-bias model?机器学习困难essay未尝试面试订阅2412Why Model Rankings Can Flip as n GrowsWhy can a simple model beat a flexible one at small n and then lose badly once n is large?机器学习中等essay未尝试面试订阅2629EMA From Zero Initialization 6Let m t = beta m t-1 + (1-beta) x t with m 0=0. Derive m t as an explicit weighted sum of x 1,...,x t.机器学习中等derivation未尝试免费2744First-Price Equilibrium Under a Quadratic CDFTwo bidders have i.i.d. private values on [0,1] with CDF F(v)=v 2. In a first-price auction, derive the symmetric equilibrium bid function and compute the bid for value 0.6.脑筋急转弯中等derivation未尝试面试订阅2746Truthful Bidding Under a Capacity PenaltyIn a second-price auction for one extra inventory slot, paying above your true value v can backfire because if you win while your realized usage is low, you incur a deterministic penalty k. If the penalty is independent of the auction price and only applies when you win, what is the effective value that should replace v in the usual Vickrey truth-telling argument?脑筋急转弯困难derivation未尝试面试订阅2751Entry Decision Before a Second-Price AuctionA bidder with private value v \in [0,1] considers entering a second-price auction against one rival who is known to enter. Values are i.i.d. Uniform (0,1), and entering costs a fixed fee c=0.08. If bidding is truthful after entry, for which values v should the bidder enter?脑筋急转弯中等derivation未尝试面试订阅2755Expected Spend in an All-Pay ContestIn the two-player all-pay auction above, suppose the prize value is V=120. Compute (a) each player's expected bid, (b) the expected highest bid, and (c) the expected total spend.脑筋急转弯中等derivation未尝试面试订阅2756Bidder Utility Under Revenue EquivalenceA bidder has value 0.9 in a two-bidder private-value environment with values i.i.d. Uniform (0,1). Compute that bidder's expected utility in (a) the first-price equilibrium and (b) the truthful second-price auction. Verify that the utilities agree.脑筋急转弯中等derivation未尝试面试订阅2763Posted Price Versus Competitive AuctionA seller can either (i) post a take-it-or-leave-it price to a single buyer with value Uniform (0,1), or (ii) run a no-reserve second-price auction between two buyers with i.i.d. values Uniform (0,1). Compute the seller's optimal expected revenue in each mechanism and compare them.脑筋急转弯中等derivation未尝试面试订阅2791Average Radius in the Unit DiskA point is drawn uniformly from the unit disk. What is the expected distance from the point to the center?脑筋急转弯中等derivation未尝试面试订阅2792Right-Hand Circular Segment ProbabilityA point is drawn uniformly from the unit disk. What is the probability that its x-coordinate exceeds 1/2?脑筋急转弯困难derivation未尝试面试订阅2793Expected Chord Length on the Unit CircleTwo points are chosen independently and uniformly on the unit circle. What is the expected length of the chord connecting them?脑筋急转弯中等derivation未尝试面试订阅2797Expected Squared Distance in the Unit DiskTwo points are chosen independently and uniformly from the unit disk. What is the expected squared Euclidean distance between them?脑筋急转弯中等derivation未尝试面试订阅2798Average Manhattan Distance in the Unit SquareTwo points are chosen independently and uniformly in the unit square. What is their expected Manhattan distance |X 1-X 2|+|Y 1-Y 2|?脑筋急转弯中等derivation未尝试面试订阅2800Average Distance to a Side of an Equilateral TriangleA point is chosen uniformly inside an equilateral triangle of side length 2. What is its expected distance to a fixed side?脑筋急转弯中等derivation未尝试面试订阅