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1590Cross-Term Budget After Eliminating One Coordinate 5For q(x,y) = 5x 2 + 2bxy + 4y 2, what is the largest absolute value |b| such that, after optimizing out y, the remaining x 2 coefficient is at least 3.75?数学困难derivation未尝试面试订阅1591Recover a Cholesky Pivot 1A positive definite matrix A = [[9, 6], [6, 13]] has Cholesky factor L = [[l11, 0], [l21, l22]] with positive diagonal. What is l22?数学简单数值题未尝试免费1592Recover a Missing Entry from a Cholesky Margin 2A matrix A = [[4, 2], [2, t]] has a Cholesky factor with positive diagonal entries 2 and 3. What is t?数学中等derivation未尝试免费1593Choose a Diagonal Entry to Hit a Cholesky Pivot 3For q(x,y) = 5x 2 + 4xy + ky 2, the Cholesky factor of the associated matrix is required to have second diagonal entry equal to 2. What value of k achieves that?数学中等derivation未尝试免费1594Residual Coefficient After Completing the Square 4Rewrite 8x 2 + 12xy + 7y 2 in the form 8(x + 3y/4) 2 + cy 2. What is c?数学困难数值题未尝试面试订阅1595Coupling Consistent With a Target Cholesky Pivot 5A matrix A = [[9, k], [k, 10]] has a Cholesky factor whose second diagonal entry is exactly 1. What values of k are possible?数学困难derivation未尝试面试订阅1596Linear Term Implied by a Target Optimizer 1For f(x) = 0.5 x T A x - b T x with A = [[4,1],[1,3]], suppose the optimizer is x* = (1,2). What vector b makes that true?数学简单derivation未尝试免费1598Second Coordinate of a Coupled Quadratic Optimizer 3For f(x) = 0.5 x T A x - b T x with A = [[6,-2],[-2,4]] and b = (8,0), what is the optimizer's second coordinate?数学中等derivation未尝试免费1599Optimizer of a Coupled Positive-Definite Penalty 4Minimize f(x,y) = 0.5(3x 2 + 4xy + 5y 2) - (6x + 8y). What is the optimizer?数学中等数值题未尝试免费1600Coupling Needed for Equal Optimizer Coordinates 5For f(x) = 0.5 x T A(t) x - b T x with A(t) = [[5,t],[t,4]] and b = (6,2), for what t does the optimizer satisfy x1 = x2?数学困难derivation未尝试面试订阅1601Why Every Covariance Matrix Is PSDWhy must every valid covariance matrix be positive semidefinite?数学简单essay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