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1435Logarithmic Mixed Expansion 5Compute lim (x->0) [ln(1 + 5x + 2x 2) - 5x] / x 2.数学困难数值题未尝试面试订阅1438Sequence Limit From a Square RootCompute lim n->∞ n[(1+2/n) (1/2) - 1].数学中等数值题未尝试免费1439Logarithmic Sequence CorrectionCompute lim n->∞ n 2 [ln(1+1/n) - 1/n].数学困难数值题未尝试面试订阅1440Square-Root Remainder 5Compute lim (x->0) [sqrt(1 + 3x + -4x 2) - 1 - (3/2)x] / x 2.数学困难数值题未尝试面试订阅1443Arcsine Cubic RemainderCompute lim x->0 [arcsin x - x] / x 3.数学中等数值题未尝试免费1445Product-Log CancellationCompute lim x->0 [ (1+x)ln(1+x) - x ] / x 2.数学困难数值题未尝试面试订阅1448Mixed-Expansion Limit 3Compute lim (x->0) [ln(1+2x) - 2 arctan(x)] / x 2.数学中等数值题未尝试免费1450Mixed-Expansion Limit 5Compute lim (x->0) [ln(cosh(2x))] / x 2.数学困难数值题未尝试面试订阅1543Sherman-Morrison Vector Update C 3Suppose A is invertible, A (-1)b = [5, 0] T, A (-1)u = [2, 1] T, v T A (-1)b = 1, and v T A (-1)u = -1/2. What is (A + u v T) (-1) b?数学中等essay未尝试免费1544Sherman-Morrison Vector Update D 4Suppose A is invertible, A (-1)b = [1, 3] T, A (-1)u = [0, 2] T, v T A (-1)b = 4, and v T A (-1)u = 2/3. What is (A + u v T) (-1) b?数学困难数值题未尝试面试订阅1545Sherman-Morrison Failure TestWhy does the Sherman-Morrison formula break down exactly when 1 + v T A (-1)u = 0?数学困难数值题未尝试面试订阅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1607Why Sample Covariance Can Be Rank DeficientWhy can a sample covariance matrix become rank deficient when the number of observations is smaller than the number of assets?数学中等essay未尝试免费1614One-Factor Covariance Entry 1Under a one-factor model X = bF + epsilon with factor variance 3, asset loadings (1, 2), and idiosyncratic variances (4, 9), what are Var(X 1), Var(X 2), and Cov(X 1, X 2)?数学困难derivation未尝试面试订阅1615Signed-Factor Covariance Entry 2Under a one-factor model X = bF + epsilon with factor variance 5, asset loadings (2, -1), and idiosyncratic variances (1, 4), what are Var(X 1), Var(X 2), and Cov(X 1, X 2)?数学困难derivation未尝试面试订阅1627Sparse Symmetric Shock Model From Variance and Fourth MomentA symmetric shock variable takes values -a, 0, and a with probabilities p, 1-2p, and p. If the sample second and fourth raw moments are m 2 and m 4, solve for a and p by method of moments.统计中等derivation未尝试免费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1634Inferring Cross-Day Heterogeneity from Paired Signal OutcomesSuppose each trading day has an unobserved hit probability P \sim Beta ( , ). Conditional on P, two independent intraday signals H 1 and H 2 are Bernoulli(P). From data, you estimate E[H 1] = 0.60, \qquad P(H 1=1, H 2=1) = 0.42. Use the method of moments to estimate and .统计中等derivation未尝试面试订阅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