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634Terminal-Variable Projection 4Let X 1, X 2, X 3, X 4 be iid symmetric ±1 variables with natural filtration F n. Define Y = X 1+X 2+X 3+X 4 and M n = E[Y | F n]. Is (M n) a martingale?概率中等derivation未尝试免费635Terminal-Variable Projection 5Let X 1, X 2, X 3, X 4 be iid symmetric ±1 variables with natural filtration F n. Define Y = (X 1+X 2+X 3) 2 and M n = E[Y | F n]. Is (M n) a martingale?概率困难derivation未尝试面试订阅636Normalized Product Process 1Let Y 1, Y 2, ... be iid positive random variables taking values [1, 3] with probabilities ['1/2', '1/2'], and let F n = sigma(Y 1,...,Y n). Define M n = (Y 1...Y n)/(2) n. Is (M n) a martingale?概率简单derivation未尝试免费638Normalized Product Process 3Let Y 1, Y 2, ... be iid positive random variables taking values [1, 4] with probabilities ['3/4', '1/4'], and let F n = sigma(Y 1,...,Y n). Define M n = (Y 1...Y n)/(7/4) n. Is (M n) a martingale?概率中等derivation未尝试免费641Martingale Diagnosis Counterexample 1M n = S n/(n+1), where S n = X 1+...+X n for iid symmetric ±1 increments. Is (M n) a martingale?概率简单derivation未尝试免费642Martingale Diagnosis Counterexample 2M n = X (n+1)-p for iid Bernoulli(p) variables with natural filtration F n. Is (M n) a martingale?概率中等derivation未尝试免费643Martingale Diagnosis Counterexample 3M n = S n 2 - n/2 for a symmetric ±1 random walk. Is (M n) a martingale?概率中等derivation未尝试免费644Martingale Diagnosis Counterexample 4M n = 2 n S n for a symmetric ±1 random walk. Is (M n) a martingale?概率困难derivation未尝试面试订阅646Predictable-Transform Martingale 1M n = sum (k=1) n S (k-1) * X k, where X k are iid symmetric ±1 and S (k-1)=X 1+...+X (k-1). Is (M n) a martingale with respect to the natural filtration?概率简单derivation未尝试免费647Predictable-Transform Martingale 2M n = sum (k=1) n (1+S (k-1) 2) * X k, where X k are iid symmetric ±1. Is (M n) a martingale with respect to the natural filtration?概率简单数值题未尝试免费648Predictable-Transform Martingale 3M n = sum (k=1) n 1 S (k-1)>0 * X k for a symmetric ±1 walk. Is (M n) a martingale with respect to the natural filtration?概率中等derivation未尝试免费649Predictable-Transform Martingale 4M n = sum (k=1) n (2+(-1) k) * X k with iid symmetric ±1 X k. Is (M n) a martingale with respect to the natural filtration?概率中等derivation未尝试免费667Biased Corridor Hit Probability 1A random walk starts at 2, moves +1 with probability 3/5 and -1 with probability 2/5, and stops when it first hits 0 or 7. What is the probability that it reaches 7 before 0?概率中等数值题未尝试免费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1667Bootstrap Bias of the Sample Maximum on a Tiny SampleA sample is 1,4 . Under the nonparametric bootstrap with resample size 2, compute the bootstrap expectation of the sample maximum and the resulting bootstrap bias estimate for the original maximum statistic.统计简单derivation未尝试免费1668Bootstrap Variance of a Bernoulli Mean From the Plug-In LawA sample of size n has empirical success rate p hat. Under the nonparametric bootstrap, what is the variance of the resampled sample mean conditional on the observed data?统计简单derivation未尝试免费1671Why the Naive Bootstrap Misses a BoundaryAn estimator is constrained to be nonnegative and lands exactly at 0 on the observed sample. Why can the naive nonparametric bootstrap badly misrepresent uncertainty near that boundary?统计简单derivation未尝试免费1672Pairs Bootstrap Versus Residual Bootstrap in Heteroskedastic DataWhy can a residual bootstrap be invalid for regression under heteroskedasticity while a pairs bootstrap remains defensible?统计中等derivation未尝试面试订阅1673When the Bootstrap Unit Should Be a Cluster, Not a RowWhy should a practitioner resample accounts, users, or securities rather than individual rows when observations inside each group share latent shocks?统计中等essay未尝试面试订阅