题目3398 · 数学
Second-Derivative Mesh From a Fourth-Derivative Bound
A centered second-derivative formula has error bounded by M h^2 / 12 with M=24. What h keeps the truncation error at or below 0.005?
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中文题目题目3212 · 概率
Second Moment of Centered Sum at a Poisson Horizon
Let $X_1,X_2,\dots$ be i.i.d. with mean $\mu$ and variance $3$. Let $N$ be independent of the increments and distributed as Poisson(4). Show that for the centered stopped sum $M_N=\sum_{i=1}^N (X_i-\mu)$, one has $E[M_N^2]$ equal to what value?
打开 →题目1598 · 数学
Second Coordinate of a Coupled Quadratic Optimizer 3
For f(x) = 0.5 x^T A x - b^T x with A = [[6,-2],[-2,4]] and b = (8,0), what is the optimizer's second coordinate?
打开 →题目406 · 概率
Second Order Statistic from Five Uniforms
Let $X_1, \ldots, X_5$ be independent $\operatorname{Uniform}(0,1)$ random variables and let $X_{(2)}$ denote the second smallest. Find $E[X_{(2)}]$.
打开 →题目3450 · 数学
Second Sensor Noise Required for 0.4 Bits Total
A latent X ~ N(0,1) is already observed by one sensor with noise variance 3. A second independent sensor with noise variance v will be added. What v makes the total mutual information 0.4 bits?
打开 →题目3392 · 数学
Second-Derivative Richardson Upgrade 2
A second-order estimate of f''(x) is 5.8 at h=0.4 and 5.2 at h=0.2. Assuming an O(h^2) truncation error, what Richardson-extrapolated estimate do you get?
打开 →题目3410 · 数学
Second-Order Forward Stencil at a Boundary
Use the second-order forward stencil (-3f(0)+4f(h)-f(2h))/(2h) with h=0.1, f(0)=1, f(0.1)=1.24, and f(0.2)=1.52. What derivative estimate do you get at 0?
打开 →题目682 · 脑筋急转弯
Sector-Liquidity Tag Collision 2
Each trade is tagged by one of 4 sectors and one of 4 liquidity buckets. Show that among 17 trades, two trades must share both tags.
打开 →题目2402 · 机器学习
Second Crossover With a Lower-Bias Flexible Model
A flexible model has excess error 0.02 + 24/n, while a simpler model has excess error 0.14 + 6/n. At what sample size do they tie?
打开 →题目3696 · 随机过程
Second Factor Drift After Neutralizing the First Carry
Under P, two processes share the same Brownian driver: dX_t = 1.2dt + 0.6dW_t and dY_t = 0.5dt + 0.25dW_t. If Q is chosen so that X has drift 0.3 under Q, what drift does Y have under Q?
打开 →题目434 · 概率
Second Failure in a Memoryless Component Array
A system has 4 independent components, each with lifetime $\operatorname{Exp}(2)$. When a component fails, it is removed and the remaining components continue operating. By memorylessness, surviving components' residual lifetimes are still $\operatorname{Exp}(2)$. Find the expect
打开 →题目2058 · 数学
Second Log-Balance Newton Step 13
The starting guess is already positive and somewhat close, so a single Newton step is informative. Apply one Newton step to solve x + 1 ln x = 2 starting from x_0 = 2.
打开 →题目353 · 概率
Second Moment of a Random Sum via the Tower Property
Let $N \sim \operatorname{Poisson}(4)$ and, given $N = n$, let $S = X_1 + \cdots + X_n$ where $X_i \stackrel{\text{iid}}{\sim} \operatorname{Uniform}(0,1)$. Use the tower property and the identity $E[S^2 \mid N] = \operatorname{Var}(S \mid N) + (E[S \mid N])^2$ to find $E[S^2]$.
打开 →题目368 · 概率
Second Moment of a Scale-Mixed Normal via Tower
Let $\Theta$ be drawn uniformly from $\{1, 2, 3\}$ and, given $\Theta = \theta$, let $X \mid \Theta{=}\theta \sim N(0, \theta)$. Using the tower property, compute $E[X^2]$ and $E[X^4]$.
打开 →题目2982 · 概率
Second Moment of a Size-Biased Piece
In the same setup as above, a random point is drawn on the stick after two cuts and $B$ is the length of the piece containing that point. Compute $E[B^2]$.
打开 →题目2066 · 数学
Second Reciprocal-Balance Newton Step 21
The reciprocal equation has an interior positive solution, and the desk wants one explicit Newton move. Apply one Newton step to solve x + 1/x = 3 starting from x_0 = 2.
打开 →题目416 · 概率
Second Smallest of Four Uniforms
Let $X_1, X_2, X_3, X_4$ be independent $\operatorname{Uniform}(0,1)$ random variables. Compute $E[X_{(2)}]$.
打开 →题目2067 · 数学
Second Two-Cashflow Yield Newton Step 22
Another bond calibration step is computed with the same Newton machinery. Solve 6/(1+y) + 106/(1+y)^2 = 101 by one Newton step starting from y_0 = 0.04.
打开 →题目5156 · 金融与交易
Second-Year Discount From a Par Coupon Bond
A 2-year annual 4% coupon bond is priced at 100. If D(1)=0.97, what is D(2)?
打开 →题目5668 · 脑筋急转弯
Secret-Santa No Self-Draw Probability
Five people draw names uniformly at random from a hat containing all five of their own names, one draw each, forming a random permutation. What is the probability that nobody draws their own name?
打开 →题目5939 · 概率
Secretary Selection When Ties Are Possible
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 →题目5940 · 概率
Secretary With an Unknown Number of Candidates
Candidates arrive one at a time in uniformly random order, but the TOTAL number N is itself random: N = 2 with probability 1/2 and N = 3 with probability 1/2, and you do not learn N in advance. After each arriving candidate you observe its rank relative to those seen so far and m
打开 →题目4397 · 机器学习
Cross-Sectional Z-Score 2
An asset returns 1.2% today while the cross-sectional mean return of its universe is 0.4%. What demeaned return feature does the asset receive?
打开 →题目4398 · 机器学习
Cross-Sectional Z-Score 3
A stock returns 1.5% while its sector index returns 0.9%. If the stock's sector beta is 1.2, what sector-residual return feature do you compute?
打开 →题目4512 · 数理金融
What to Challenge First When Someone Says 'Three Securities, Three States, So Complete'
A PM says 'there are three securities and three states, so the market is complete.' What is the first structural check you should push back with?
打开 →题目4506 · 数理金融
Why Counting Securities Is Not Enough for Completeness
A candidate says a 4-state market with 4 traded securities must be complete. Why is that claim too quick?
打开 →题目4396 · 机器学习
Cross-Sectional Z-Score 1
If today's close is 100, tomorrow's open is 98, and tomorrow's close is 99, what is the next-day open-to-close return that would be used as an intraday label available after tomorrow's session?
打开 →题目4399 · 机器学习
Cross-Sectional Z-Score 4
Yesterday's return was 1.8%. The trailing mean of completed daily returns is 0.3% and the trailing standard deviation is 0.5%. What lagged return z-score feature do you record?
打开 →题目4400 · 机器学习
Cross-Sectional Z-Score 5
A stock closes at 50 yesterday and opens at 51 today, while the market index closes at 2000 yesterday and opens at 2020 today. If the stock's overnight beta to the market is 1.5, what market-adjusted overnight return feature do you compute?
打开 →题目5860 · 数学
Overdamped Second-Order ODE with Distinct Real Roots
Solve $y''-5y'+6y=0$ with $y(0)=1$ and $y'(0)=0$.
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