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中文题目A factory produces items independently, each defective with probability $p = 0.03$. A batch of $n = 500$ items is inspected. Let $D$ denote the number of defective items in the batch. Using the Central Limit Theorem, approximate $P(D \le 20)$. You may use $\Phi(1.30) \approx 0.
打开 →Why can the midpoint rule outperform the trapezoid rule on smooth integrands even though both are second-order?
打开 →Why is purging not the same thing as embargoing in time-series validation?
打开 →Why do swaption payoffs often point directly to the annuity numeraire rather than to a single zero-coupon bond?
打开 →Why are bounded stopping times the cleanest setting for applying optional stopping in interview problems?
打开 →Why should OST problems not be confused with reflection-principle problems even though both often involve Brownian barriers?
打开 →A gambler starts with $\$3$. Each round she wins $\$1$ with probability $\tfrac{2}{3}$ or loses $\$2$ with probability $\tfrac{1}{3}$. She stops when her fortune first hits $\$0$ or below (ruin) or reaches $\$5$ or above (victory). Note that if her fortune is $\$1$ and she loses,
打开 →Consider a Markov chain on $\{0, 1, 2, \ldots\}$ with absorbing state $0$. From state $k \ge 1$, the chain moves to $k+1$ with probability $p$ and to $k-1$ with probability $q = 1 - p$, where $0 < p < 1$. However, there is a **reflecting barrier** at state $N$: from state $N$, th
打开 →An exchange gateway receives $N\sim\mathrm{Poisson}(100)$ messages in a fixed interval. Use the Poisson Chernoff upper-tail bound to estimate $P(N\ge 130)$.
打开 →Let $X_n \sim \text{Binomial}(n, \lambda/n)$ for fixed $\lambda > 0$ and $n > \lambda$. (a) Write down the characteristic function $\varphi_{X_n}(t) = E[e^{itX_n}]$ in closed form. (b) Show that $\lim_{n \to \infty} \varphi_{X_n}(t) = e^{\lambda(e^{it} - 1)}$ for every $t \in \
打开 →Why is the exponential martingale the natural OST tool once Brownian motion has nonzero drift?
打开 →Why does the compensated square W_t^2-t become the natural OST object for expected exit times?
打开 →Why is it dangerous to apply optional stopping mechanically when the stopping rule is unbounded?
打开 →Why does low-rank truncation often denoise a matrix instead of merely compressing it?
打开 →Why can equal-weighting many correlated alphas fail to deliver the diversification that the count of signals seems to promise?
打开 →A quality line inspects items one at a time; each item is defective independently with probability $0.05$. Inspection stops at the first defective item. Each inspection (defective or not) costs an i.i.d. amount $C_i$ with $E[C_i]=\$8$, independent of the defect outcomes. Let $N$
打开 →Why should finite-horizon barrier-touch probabilities not be confused with the interval-hitting probabilities used in bounded first-passage problems?
打开 →Let $X$ be a nonnegative integer-valued random variable with PGF $G_X(s)$. Each of the $X$ items is independently kept with probability $p$. If $Y$ is the number kept, express $G_Y(s)$ in terms of $G_X$.
打开 →Three items arrive in random order; after each you learn its rank relative to those seen so far and must accept or reject irrevocably (the last is forced). Instead of trying to get the single best, you want to minimize the expected absolute rank of the item you finally accept (ra
打开 →At the daily risk meeting, the CRO notes that the standalone historical VaRs of three desks add up to 42 million, but the portfolio historical VaR comes out to only 31 million after a new basis book was added. What is the most likely explanation, and what is one concrete systems
打开 →In one paragraph, contrast what reflection principle and optional stopping are each best at in Brownian barrier problems.
打开 →Three items arrive in uniformly random order. Their qualities are NOT all distinct: two of them have quality 2 (tied for best) and one has quality 1. After each item you observe its quality relative to those seen so far, reported as 'higher', 'tied', or 'lower' (so a tie is visib
打开 →Four items arrive in uniformly random order; you observe only relative ranks and must accept one irrevocably (if you reach the last item you take it). Unlike the classic problem, you win if the item you accept is either the best OR the second-best overall. Find the policy that ma
打开 →Why do desks clean or smooth option surfaces before using them in local-vol or risk systems?
打开 →A trader must ferry three items — a fox, a goose, and a sack of beans — across a river in a small boat that can carry the trader plus at most one item per trip. If left together unattended, the fox eats the goose, and the goose eats the beans (the fox ignores the beans). The trad
打开 →In one-defective search problems, why does adding the possibility of no defect change the counting argument in a substantive way?
打开 →How can annuity, bond, and money-market numeraires all be valid choices for related rates problems without creating contradiction?
打开 →If a nontraded claim seems to have several candidate prices, what should you verify before arguing about which probability measure is 'correct'?
打开 →Why can a larger forest fail to repair performance when the training labels themselves are systematically corrupted?
打开 →If the terminal payoff is already specified, why do boundary conditions still matter in many Feynman-Kac problems?
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