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194Variance of the Number of Occupied UrnsFour distinguishable balls are thrown independently and uniformly at random into 3 distinguishable urns. Let N be the number of nonempty urns. Find Var (N). Give an exact fraction.概率困难数值题未尝试免费334Robust Covariance of Overlapping Sums of Independent VariablesLet X, Y, Z be independent random variables with Var (X) = 1, Var (Y) = 2, and Var (Z) = 3. Define U = X + Y and V = Y + Z. Compute Cov (U, V) and Corr (U, V). 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1602Equal-Weight Two-Asset Variance 1Two assets have variances 4 and 9, with covariance 2. What is the variance of the portfolio with weights (1/2, 1/2)?数学简单数值题未尝试免费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1618Negative Correlation Extraction 2Two assets have variances 4 and 25, and covariance -6. What is their correlation?数学中等derivation未尝试免费1797Signal Stationarity Classification 2A candidate signal is defined by X t = 0.2 t + ε t. Is it weakly stationary?统计中等derivation未尝试免费1798Signal Stationarity Classification 3A candidate signal is defined by X t = ε t + s t where s t is a fixed deterministic day-of-week pattern. Is it weakly stationary?统计简单derivation未尝试免费1818Why Ergodicity MattersWhy is ergodicity stronger than stationarity, and why do practitioners care about it when they average one long signal history?统计中等derivation未尝试免费1822AR(1) Multi-Step Forecast 2A signal follows X t = 4 + 0.7 X (t-1) + e t with Var(e t) = 1.5 and current value X t = 8. What is the h = 2 step forecast E[X (t+2) | X t]?统计简单derivation未尝试免费1828MA(1) Lag-1 Correlation 3A microstructure noise model uses Y t = e t + -0.4 e (t-1). What is its lag-1 autocorrelation rho(1)?统计简单数值题未尝试免费1843ARMA Identification or Simplification 3You observe the diagnostic statement: Both ACF and PACF tail off. What is the correct modeling conclusion?统计中等derivation未尝试面试订阅3277Covariance of PCA ScoresIf data are centered and projected onto orthonormal eigenvectors of the covariance matrix, what is the covariance matrix of the resulting PCA score vector?数学中等derivation未尝试面试订阅3284Why Centering Matters Before PCAWhy can skipping mean-centering make the first principal component mostly track the mean level instead of genuine variation?数学中等essay未尝试面试订阅4016Minimum-Variance Hedge Ratio 1A spot exposure has volatility 18%, the hedging futures has volatility 24%, and the desired minimum-variance hedge ratio is 0.6. What correlation is implied by h*=rho*sigma S/sigma F?金融与交易中等derivation未尝试面试订阅4017Minimum-Variance Hedge Ratio 2A spot exposure has volatility 30%, correlation 0.75 to the hedging futures, and desired hedge ratio 0.9. What futures volatility is implied?金融与交易中等derivation未尝试面试订阅4018Minimum-Variance Hedge Ratio 3A short hedge ratio of -0.3 is desired. If futures volatility is 20% and correlation is -0.4, what spot volatility is implied?金融与交易中等derivation未尝试面试订阅4021Hedge Contract Count 1A spot exposure of 2000 units is hedged with futures contracts of 100 units each. Spot volatility is 30.00%, futures volatility is 25.00%, and correlation is 0.8. How many futures contracts correspond to the minimum-variance hedge?金融与交易中等derivation未尝试面试订阅4031Residual Hedge Variance 1Spot volatility is 20% and correlation to the hedging futures is 0.8. Under the minimum-variance hedge, what residual spot volatility remains?金融与交易中等derivation未尝试面试订阅