题目2620 · 机器学习
A Bound on Total Function Movement 8
Suppose every boosting round changes any one point's prediction by at most eta A in absolute value. What upper bound does this imply on the total prediction movement after M rounds?
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中文题目题目3221 · 统计
A Confidence Interval Cannot Price a One-Off Launch Decision
A PM sees a frequentist 95% confidence interval for next-month strategy edge of [-0.1, 0.4] and asks, "So what is the probability the true edge is positive for this launch decision?" Why can't the interval answer that question by itself, and what Bayesian quantity would answer it
打开 →题目5827 · 金融与交易
A dealer faces order flow where a fraction 0.4 of traders ar
A dealer faces order flow where a fraction 0.4 of traders are informed (always trade in the correct direction) and 0.6 are noise traders who buy or sell with equal probability. The true value is equally likely high or low, so informed traders buy half the time and sell half the t
打开 →题目4165 · 机器学习
A Fast Sanity Check for Gen-vs-Disc Questions
The research desk wants not only labels but also synthetic feature draws conditional on each class for stress testing. Which side is the more natural starting point?
打开 →题目2736 · 脑筋急转弯
A Fibonacci Binomial Sum From Coefficient Extraction
Evaluate sum_{k>=0} C(12-k, k). You should explain why a generating function turns this into a Fibonacci-type coefficient.
打开 →题目063 · 概率
A Function of Independent Variables Need Not Be Independent of Its Inputs
Let $X$ and $Y$ be independent $\text{Bernoulli}(1/2)$ random variables. Define $W = \max(X, Y)$. (a) Compute the distribution of $W$. (b) Determine whether $X$ and $W$ are independent by checking all four joint probabilities $P(X = x, W = w)$ for $x, w \in \{0, 1\}$.
打开 →题目1653 · 统计
A Smoothed Bernoulli Estimator vs the Sample Proportion
Let $X\sim \mathrm{Binomial}(10,p)$ and consider the estimator $$\delta = \frac{X+1}{12}$$ for $p$. At the parameter value $p=0.2$, compute the bias, variance, and MSE of $\delta$, and compare its MSE with the usual sample proportion $\hat p = X/10$.
打开 →题目2887 · 概率
A/B Gap Concentration
You run an A/B test with $n$ Bernoulli observations in treatment and $n$ in control, all independent. Let $\bar X$ and $\bar Y$ be the sample means. Use Hoeffding's inequality to bound \[ P\bigl((\bar X-\bar Y)-E[\bar X-\bar Y]\ge \varepsilon\bigr). \]
打开 →题目3794 · 随机过程
Absorbing Boundary as Path Termination
How should an absorbing boundary condition be interpreted inside a Feynman-Kac representation?
打开 →题目524 · 概率
Absorption with a Long-Range Jump
A particle moves on $\{0, 1, 2, 3, 4, 5\}$ with absorbing states $0$ and $5$. The transitions from transient states are: - From $1$: to $0$ w.p. $1/2$, to $2$ w.p. $1/2$. - From $2$: to $1$ w.p. $1/3$, to $3$ w.p. $1/3$, to $4$ w.p. $1/3$ (a long-range jump). - From $3$: to $2$ w
打开 →题目398 · 概率
Additivity of Chi-Squared Distributions via MGF
Let $X \sim \chi^2(m)$ and $Y \sim \chi^2(n)$ be independent. Using moment-generating functions, prove that $X + Y \sim \chi^2(m + n)$.
打开 →题目3404 · 数学
Advection-Diffusion Explicit Time-Step Rule 1
Use the simple sufficient rule Delta t <= 1 / (|u|/Delta x + 2 kappa / Delta x^2) for an explicit advection-diffusion update. If u = 1, kappa = 0.2, and Delta x = 0.1, what is the resulting Delta t cap?
打开 →题目2076 · 数理金融
Affine Replication Check in an Incomplete Market 6
The stock is 100 today and ends at 120, 100, or 80 next period. Consider a claim paying 20, 10, and 0 in those three states. Can it be replicated exactly using only the stock and cash? If yes, give the hedge. If not, identify the replication obstruction.
打开 →题目3162 · 统计
Allocate Capital Only Above a 60% Posterior
A latent profitable regime has prior probability $\frac{2}{5}$. Independent signals arrive sequentially; each `H` doubles the odds of the regime and each `T` halves them. After the signal string `HHT`, should you act if the required posterior threshold is $\frac{3}{5}$? Also repo
打开 →题目4940 · 数理金融
Allocation Intuition 25
Why is a single firm-wide VaR or ES number not enough for desk incentives unless it is broken into component contributions?
打开 →题目2940 · 概率
Almost Sure Extinction in Every Subcritical Process
A Galton-Watson process has offspring mean $m<1$. What is its extinction probability?
打开 →题目2923 · 概率
Another Exact Extinction Root
Each individual has offspring distribution \[ P(\xi=0)=0.2,\qquad P(\xi=1)=0.5,\qquad P(\xi=2)=0.3. \] Starting from one ancestor, compute the extinction probability.
打开 →题目1689 · 统计
Approximate Fair-Coin Test with a Normal Approximation
A coin is flipped 20 times and lands heads 14 times. Use the normal approximation to test fairness at the 5% two-sided level.
打开 →题目5181 · 金融与交易
Arbitrage Direction 1
Spot is 100, maturity is 1 years, and the funding rate is 0.03. The quoted forward price is 104. Assuming no income and no frictions, which arbitrage direction is indicated, and what is the mispricing per unit relative to fair value?
打开 →题目5182 · 金融与交易
Arbitrage Direction 2
Spot is 90, maturity is 0.5 years, and the funding rate is 0.04. The quoted forward price is 91. Assuming no income and no frictions, which arbitrage direction is indicated, and what is the mispricing per unit relative to fair value?
打开 →题目1841 · 统计
ARMA Identification or Simplification 1
You observe the diagnostic statement: ACF tails off geometrically, PACF cuts after lag 1. What is the correct modeling conclusion?
打开 →题目1842 · 统计
ARMA Identification or Simplification 2
You observe the diagnostic statement: ACF cuts after lag 1, PACF tails off. What is the correct modeling conclusion?
打开 →题目1843 · 统计
ARMA Identification or Simplification 3
You observe the diagnostic statement: Both ACF and PACF tail off. What is the correct modeling conclusion?
打开 →题目1844 · 统计
ARMA Identification or Simplification 4
You observe the diagnostic statement: AIC prefers ARMA(2,1) but BIC prefers ARMA(1,1). What is the correct modeling conclusion?
打开 →题目1845 · 统计
ARMA Identification or Simplification 5
You observe the diagnostic statement: (1-0.5L) X_t = (1-0.5L) e_t. What is the correct modeling conclusion?
打开 →题目5862 · 数理金融
Arrow-Debreu Prices From Option Quotes
A stock has three future states with prices 120, 100, and 80; the risk-free rate is 0. Calls struck at 80 trade at 28 and calls struck at 100 trade at 8. Using the digital/butterfly decomposition, find the Arrow-Debreu price of the single highest state (the state where the stock
打开 →题目4655 · 数理金融
Assumption Breakdown Diagnostic 15
Before blaming every option-pricing error on volatility misspecification, what should you ask first?
打开 →题目4656 · 数理金融
Assumption Breakdown Diagnostic 16
If you hedge more frequently, what usually happens to diffusion-style replication error and what usually happens to transaction costs?
打开 →题目4658 · 数理金融
Assumption Breakdown Diagnostic 18
If vol-of-vol rises materially, what usually happens to the plausibility of a single constant-volatility Black-Scholes description?
打开 →题目4659 · 数理金融
Assumption Breakdown Diagnostic 19
If market liquidity deteriorates sharply, which Black-Scholes assumption becomes more dangerous to ignore?
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