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1836AR(1) Forecast Error Variance 1For the AR(1) model X t = phi X (t-1) + e t with phi = 0.6 and Var(e t) = 1, what is the h = 3 step forecast error variance?统计简单derivation未尝试免费1841ARMA Identification or Simplification 1You observe the diagnostic statement: ACF tails off geometrically, PACF cuts after lag 1. What is the correct modeling conclusion?统计简单数值题未尝试免费1844ARMA Identification or Simplification 4You observe the diagnostic statement: AIC prefers ARMA(2,1) but BIC prefers ARMA(1,1). What is the correct modeling conclusion?统计简单derivation未尝试免费1906Minimum Embargo Length 1A label is defined as the forward return over the next 5 trading days. If you split train and test windows by time, what is the minimum embargo gap needed to avoid leaking label information across the boundary?统计简单essay未尝试免费1911False Launch Probability 1You test 40 independent noise strategies. A strategy is launched only if it passes an in-sample screen at level 1/20 and then passes a second independent validation at level 1/10. Under the global null, what is the probability that at least one noise strategy still gets launched?统计简单essay未尝试免费1916Look-Ahead Audit Scenario 1A daily factor is z-scored using the full-sample mean and standard deviation before the backtest is run. What is the main problem?统计简单essay未尝试免费1926Minimum Detectable Edge 1Using the same approximation n >= K * sigma 2 / delta 2, suppose K = 7.84, sigma = 25 bp, and you can collect only n = 400 observations. What is the smallest detectable mean edge delta, in basis points?统计简单essay未尝试免费1931Effective Sample Under Serial Dependence 1A return series has 240 observations and lag-1 autocorrelation 1/5. Using the heuristic n eff = n * (1-rho)/(1+rho), what is the effective sample size?统计简单essay未尝试免费1936Paired Design Observation Ratio 1Two strategies are tested on the same names and dates, giving correlation 1/5 between their noise terms. Relative to an unpaired comparison with the same marginal variance, by what factor does the paired design reduce required sample size?统计简单数值题未尝试免费1941Power Score Design Choice 1Compare two research designs using the score delta * sqrt(n) / sigma. Design A has delta=4 bp, n=225, sigma=2 bp. Design B has delta=3 bp, n=400, sigma=1 bp. Which design has the larger score?统计简单数值题未尝试免费1946Log Utility Versus Impact Budget 1A signal is strong enough that the PM scores larger size through a log benefit but pays quadratic impact. The desk chooses a scalar exposure x > -1 to maximize F(x) = 3 ln(1+x) - 2 x 2. What exposure x maximizes F?数学简单数值题未尝试免费1947Log Utility Versus Impact Budget 2A desk chooses a participation tilt with diminishing marginal benefit and quadratic implementation drag. The desk chooses a scalar exposure x > -1 to maximize F(x) = 4 ln(1+x) - 1 x 2. What exposure x maximizes F?数学简单数值题未尝试免费1948Log Utility Versus Impact Budget 3A portfolio manager sizes one smooth alpha sleeve with a log reward and a quadratic risk surcharge. The desk chooses a scalar exposure x > -1 to maximize F(x) = 12 ln(1+x) - 1 x 2. What exposure x maximizes F?数学中等数值题未尝试免费1949Derive the Optimizer for Log Utility Minus Quadratic Cost 4For parameters a>0 and b>0, derive the unique maximizer of F(x)=a ln(1+x)-b x 2 on x>-1.数学中等derivation未尝试免费1951Bid-Ask Skew Balance 6On one quote axis, the maker gets more value from aggressive bids than from aggressive offers. A market maker chooses a skew x in (-1,1) to maximize G(x) = 5 ln(1+x) + 3 ln(1-x). What skew is optimal?数学简单数值题未尝试免费1952Derive the Optimizer for an Asymmetric Log Skew Objective 7For a>0 and b>0, derive the unique maximizer of G(x)=a ln(1+x)+b ln(1-x) on x in (-1,1).数学中等derivation未尝试免费1956Carry Drag Versus Quadratic Inventory 11A carry term explodes as the state approaches -1, so the desk cannot simply push x downward. Minimize H(x) = 4 x 2 + 9/(1+x) over x > -1.数学简单数值题未尝试免费1957Funding Buffer Versus Risk Penalty 12A funding buffer term falls with x while the quadratic penalty rises, creating an interior optimum. Minimize H(x) = 3 x 2 + 24/(1+x) over x > -1.数学简单数值题未尝试免费1958Verify the Designed Minimizer in a Quadratic-Reciprocal Objective 13Let r>0 and a>0. Show that x=r uniquely minimizes H(x)=a x 2 + 2 a r(1+r) 2 /(1+x) over x>-1.数学中等derivation未尝试免费1961Exponential Imbalance Optimizer 16A smooth response score is J(x)=e x + 4 e -2x . What x minimizes J?数学简单数值题未尝试免费