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中文题目A model starts at validation accuracy 0.82. Permuting feature X1 alone lowers it to 0.79, permuting X2 alone lowers it to 0.8, and permuting them together lowers it to 0.7. What are the three drops, and what pattern does that suggest?
打开 →A model has baseline AUC 0.70. With a correlated twin present, permuting feature A drops AUC to 0.64. After removing the twin, permuting A drops AUC to 0.58. By how much did feature A's permutation importance increase?
打开 →A feature's permutation importance is measured as baseline accuracy minus permuted accuracy. Under validation set A the numbers are 0.80 and 0.78; under noisier validation set B they are 0.74 and 0.72. What is the relative drop as a percentage of baseline in each case?
打开 →If several sector dummy variables move together and share the same economic information, what diagnostic is often better than permuting one dummy at a time?
打开 →On a two-step recombining tree with spot=64, strike=70, u=1.25, d=0.8, r=0, Δt=1, price the European call by weighting the three terminal payoffs with the binomial probabilities q^2, 2q(1-q), (1-q)^2.
打开 →An impurity-based feature ranking is id_hash=0.40, signal_1=0.35, signal_2=0.25. After limiting max depth, id_hash gain is cut in half while the other raw gains are unchanged. What are the new normalized shares?
打开 →A PDE contains u_t + L u - r u = 0 with terminal condition u(T,x)=g(x). What discounting structure should appear in the Feynman-Kac expectation, and why?
打开 →Two fixed-strike lookback calls share the same strike and terminal spot. Path A reached a higher running maximum than Path B before expiry. Which call has the larger payoff?
打开 →A permutation $\sigma$ of $\{1,2,3,4\}$ is chosen uniformly at random (each of the $4! = 24$ permutations equally likely). For $i = 1,2,3,4$, define the event $A_i = \{\sigma(i) = i\}$ (element $i$ is a fixed point). (a) By counting, show that $P(A_i) = 1/4$ for every $i$, and $P
打开 →A weak feature shows slightly negative permutation importance on a finite validation set. Should you immediately conclude it is genuinely anti-predictive?
打开 →Let $\sigma$ be a uniformly random permutation of $\{1, 2, \dots, n\}$. An ascent at position $i$ (for $1 \le i \le n-1$) is a position where $\sigma(i) < \sigma(i+1)$. Find the expected number of ascents. Additional robustness twist: before observation, an independent random re
打开 →Let $\sigma$ be a uniformly random permutation of $\{1, 2, \dots, n\}$. Find the expected number of cycles in the cycle decomposition of $\sigma$. Express your answer as a familiar function of $n$. Additional robustness twist: before observation, an independent random relabeling
打开 →Let $\sigma$ be a permutation of $\{1, 2, \dots, n\}$ chosen uniformly at random. A descent is a position $i \in \{1, \dots, n-1\}$ where $\sigma(i) > \sigma(i+1)$. What is the expected number of descents? Additional robustness twist: before observation, an independent random re
打开 →Let $\sigma$ be a uniformly random permutation of $\{1, 2, \dots, n\}$. Call position $i \in \{2, \dots, n-1\}$ an intermediate position if $\sigma(i)$ is strictly between $\sigma(i-1)$ and $\sigma(i+1)$, i.e., $\min(\sigma(i-1), \sigma(i+1)) < \sigma(i) < \max(\sigma(i-1), \sigm
打开 →If the expectation contains an accumulated running reward integral_t^T f(s,X_s) ds added to the terminal payoff, what sign should f have inside the PDE representation?
打开 →Why does the Sherman-Morrison formula break down exactly when 1 + v^T A^(-1)u = 0?
打开 →Classify $X_t=\int_0^t W_s\,dW_s+0.2t$ as a martingale, submartingale, or supermartingale.
打开 →Why should the terminal condition u(T,x)=g(x) be read as the payoff rule at the horizon rather than as an arbitrary boundary artifact?
打开 →If the terminal payoff is already specified, why do boundary conditions still matter in many Feynman-Kac problems?
打开 →Why do compensation terms like $-t$ or $- frac12 a^2 t$ appear so often when building martingales from Brownian motion?
打开 →Why do smooth finite-variation terms fail to contribute to quadratic variation even though they can dominate the path in level?
打开 →Why does the derivative in a bond-yield Newton step come with negative discounted cash-flow terms?
打开 →How should an absorbing boundary condition be interpreted inside a Feynman-Kac representation?
打开 →Suppose each individual has one child with probability $p$ and zero children with probability $1-p$. What is the extinction probability?
打开 →What is det(5 I + -1 11^T) as an n=3 dimensional matrix?
打开 →Apply It\^o to $X_t=W_t/(1+t)$. What is $dX_t$?
打开 →Show that H(w_1,w_2)=2w_1^2+5w_2^2+3(w_1+w_2)^2 is convex.
打开 →For 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?
打开 →For q(x,y) = 4x^2 + 2kxy + 7y^2, what is the largest absolute value |k| such that q(x,y) >= 2(x^2 + y^2) for all (x,y)?
打开 →An invertible matrix A has det(A) = 12. For vectors u and v, suppose v^T A^(-1) u = 1/2. What is det(A + u v^T)?
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