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1605Hedged Pair Variance 4Two assets have variances 1 and 9, with covariance -1. What is the variance of the portfolio with weights (1/3, 2/3)?数学困难数值题未尝试面试订阅1608Minimum-Variance Hedge Ratio 1You hedge X with h units of Y and want to minimize Var(X - hY). If Cov(X, Y) = 8 and Var(Y) = 4, what is the optimal hedge ratio h?数学中等derivation未尝试免费1611Equicorrelation Validity Threshold 1For an n=4 equicorrelation matrix with 1s on the diagonal and rho off the diagonal, what is the lowest rho that still keeps the matrix positive semidefinite?数学简单数值题未尝试免费1616How to Read Diagonal and Off-Diagonal EntriesIn a covariance matrix, what do the diagonal entries and off-diagonal entries mean?数学简单essay未尝试免费1617Correlation from Covariance 1Two assets have variances 9 and 16, and covariance 6. What is their correlation?数学中等derivation未尝试免费1619Why Large Condition Number Destabilizes WeightsWhy does a covariance matrix with a very large condition number make optimized portfolio weights unstable?数学困难essay未尝试面试订阅1626Recovering a Three-Point Support Scale From Two MomentsA random variable takes values 0, a, and 3a with probabilities 1-2p, p, and p. If the empirical first two raw moments are m 1 and m 2, solve for a and p by method of moments.统计简单derivation未尝试免费1628Estimating a Zero-Inflated Order-Arrival ModelPer-second order arrivals are modeled as follows: with probability the market is inactive and the observed count is exactly 0; with probability 1- , the count is Poisson ( ). From data, the empirical zero frequency is 0.70 and the empirical mean count is 0.60. Use the method of moments to estimate ( , ).统计困难derivation未尝试面试订阅1629MoM for a Random Amplitude Bernoulli CountLet X=AZ where Z is Bernoulli with success probability p and the success amplitude A is a positive constant. If the sample mean is m 1 and the sample second raw moment is m 2, solve for A and p.统计中等derivation未尝试免费1630Shifted Exponential Calibration from Raw MomentsA toy latency model assumes X = c + Y, \qquad Y \sim Exp ( ), with unknown deterministic floor c>0 and unknown rate . From historical data, the empirical mean of X is 8 and the empirical second raw moment is 73. Use the method of moments to estimate c and .统计简单derivation未尝试免费1632Estimating Activity and Size in a Zero-Inflated Fill ModelConsider a toy fill-size model for a child order. With probability 1-p, no fill occurs and the observed size is 0. With probability p, a fill occurs and the size is exponentially distributed with rate . The empirical mean fill size is 2 and the empirical variance is 12. Use the method of moments to estimate p and .统计困难derivation未尝试面试订阅1635Estimating a Three-Point Shock Model from Even MomentsA stylized inventory-shock model assumes the one-step PnL jump X takes values -a, 0, and +a with probabilities p/2, 1-p, and p/2, respectively, where a>0 is unknown. From data, the empirical second moment is 2 and the empirical fourth moment is 10. Use the method of moments to estimate p and a.统计简单derivation未尝试免费1636MLE of a Bernoulli Signal Hit RateA binary trading signal was profitable on 44 of the last 80 trading days. Model each day as an independent Bernoulli(p) outcome. Find the maximum likelihood estimator of p, and then estimate the probability that the next 3 days are all profitable under the fitted model.统计简单derivation未尝试免费1637MLE of a Poisson Order-Arrival IntensityDuring a 40-minute observation window, a venue records 120 child-order arrivals. Model the arrivals as a homogeneous Poisson process with intensity arrivals per minute. Find the MLE of , and estimate the probability of seeing zero arrivals in the next minute under the fitted model.统计简单derivation未尝试免费1638MLE of an Exponential Waiting-Time ModelTen independent waiting times between mid-price changes sum to 25 seconds. Model each waiting time as Exp ( ). Find the MLE of , and under the fitted model compute the median waiting time.统计简单derivation未尝试免费1639Joint MLEs in a Normal ModelSuppose X 1,\dots,X 9 are modeled as i.i.d. N( , 2). From the sample you know that X = 5, \qquad \sum i=1 9 (X i- X) 2 = 18. Find the MLEs of and 2.统计简单derivation未尝试面试订阅1640MLE of a Uniform Upper BoundFive i.i.d. observations are modeled as Uniform (0, ). The sample maximum is 7.4. Find the MLE of , and then estimate the median of the fitted distribution.统计简单derivation未尝试面试订阅1641MLE of a Geometric Success ProbabilityA strategy is repeatedly tried until the first profitable fill. Let X be the number of attempts until the first success, with support 1,2,\ldots, and model X\sim Geometric (p). If the sample mean from many independent episodes is 4, find the MLE of p. Under the fitted model, what is the probability that at least 4 attempts are needed?统计中等derivation未尝试面试订阅1642MLE in a Three-State Multinomial ModelA market regime model has three states: calm, trending, and dislocated, with probabilities (p 1,p 2,p 3). Over 100 days, the observed counts are 20 calm days, 30 trending days, and 50 dislocated days. Find the MLE of (p 1,p 2,p 3).统计简单derivation未尝试免费1643MLE of a Pareto Tail IndexSuppose large execution slippage magnitudes are modeled as Pareto with known scale x m=1 and unknown tail index , so the density is f(x)= x - -1 , \qquad x\ge 1. If n=8 observations satisfy \sum i=1 8 \log X i = 12, find the MLE of . Then estimate P(X>10) under the fitted model.统计中等derivation未尝试面试订阅