题目1519 · 数学
Adjusted Gross Multiplier 4
Use a second-order Taylor approximation around 0 to estimate (1+3x)^(2) * (1+1x)^(-1) at x=1/40.
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中文题目题目1995 · 数学
Alpha Maximization With Uneven Risk Units 25
The second sleeve has both larger alpha and a different risk unit, so the optimal point must balance both effects. Maximize 4x + 6y subject to 4x^2 + 9y^2 = 225.
打开 →题目1451 · 数学
Alternating-Input Recurrence 1
A sequence satisfies x_(n+1) = 1/2 x_n + c_n, where c_n = 1 on odd n and c_n = 3 on even n. Assuming any starting value x_1, what are the limits of the even and odd subsequences?
打开 →题目1523 · 数学
Asymmetric Buffer Calibration 3
Choose c so that exp(-1x + c x^2) - 1 - -1x has quadratic coefficient 4.
打开 →题目470 · 概率
Asymptotic Distribution of the Sample Median
Let $X_1, \ldots, X_n$ be i.i.d.\ $\mathrm{Uniform}(0,1)$ and let $M_n$ denote the sample median (the middle order statistic for odd $n$, or the average of the two middle values for even $n$). The asymptotic theory of order statistics gives $$\sqrt{n}\,(M_n - m) \xrightarrow{d}
打开 →题目1410 · 脑筋急转弯
ATM Call Proxy 2
Use the desk proxy premium ~= 0.4 * S * sigma * sqrt(T). If S=120, sigma=30%, and T=0.25 years, what premium estimate results?
打开 →题目1962 · 数学
Balanced Exponential Penalties 17
A desk minimizes J(x)=6 e^x + 3 e^{-2x}. What x is optimal?
打开 →题目1966 · 数学
Balanced Softplus Tradeoff 21
The linear reward and saturation penalty balance exactly at a central point. The desk maximizes K(x) = 2 x - 4 ln(1+e^x). What x is optimal?
打开 →题目2008 · 数学
Barrier Cost With Strong Quadratic Curvature 13
Both the quadratic inventory term and the capacity wall are active sources of curvature. Show that f(q) = 4 q^2 + 3/(1-q) is strictly convex on q<1.
打开 →题目1425 · 脑筋急转弯
Basis-Point Revenue 17
What is 7.5 bp of $640 million?
打开 →题目2634 · 机器学习
Batch-Average Gradient 9
If the minibatch loss is the average L = (1/B) sum_{i=1}^B L_i, derive dL/dw in terms of the per-example gradients.
打开 →题目2642 · 机器学习
BatchNorm Running Mean Update 13
A BatchNorm layer updates its running mean by mu_new = m mu_old + (1-m) mu_batch. What does this formula mean operationally?
打开 →题目454 · 概率
Berry-Esseen Bound for a Skewed Bernoulli Sum
Let $X_1, X_2, \ldots, X_n$ be i.i.d.\ $\operatorname{Bernoulli}(p)$ with $p = 0.01$ and $n = 10{,}000$. Define $S_n = \sum_{i=1}^{n} X_i$. **(a)** Using the CLT, approximate $P(S_n \le 80)$. **(b)** The Berry-Esseen theorem states that $\sup_x |P(Z_n \le x) - \Phi(x)| \le \fra
打开 →题目465 · 概率
Berry-Esseen Bound for a Sum of Uniform Random Variables
Let $U_1, \ldots, U_n$ be i.i.d.\ $\mathrm{Uniform}(0,1)$ and $S_n = \sum_{i=1}^n U_i$. The Berry-Esseen theorem states $$\sup_x \left|P\!\left(\frac{S_n - n/2}{\sigma\sqrt{n}} \le x\right) - \Phi(x)\right| \le \frac{C\,\rho}{\sigma^3 \sqrt{n}},$$ where $\sigma^2 = \mathrm{Var}(U
打开 →题目1951 · 数学
Bid-Ask Skew Balance 6
On one quote axis, the maker gets more value from aggressive bids than from aggressive offers. A market maker chooses a skew x in (-1,1) to maximize G(x) = 5 ln(1+x) + 3 ln(1-x). What skew is optimal?
打开 →题目153 · 概率
Birthday Collision Probability and Exponential Approximation
Fifty people are in a room. Each birthday is independent and uniform on $\{1, 2, \ldots, 365\}$. (a) Write the exact probability that at least two share a birthday. (b) Derive a useful upper bound on $P(\text{all distinct})$ using the inequality $1-x \le e^{-x}$ and simplify. H
打开 →题目1405 · 脑筋急转弯
Block Trade Gross 5
A desk proxy for gross block-trade dollars is block count * average block size in dollars. If those inputs are 1250 and $14 million, what estimate results?
打开 →题目3408 · 数学
Boundary Curvature From a Nonzero Slope Condition
Suppose u(0)=4, u_1=5, h=0.25, and the left-boundary slope satisfies u_x(0)=-2 approximated by (u_1-u_{-1})/(2h). What second derivative estimate do you get from (u_{-1} - 2u_0 + u_1)/h^2?
打开 →题目3409 · 数学
Boundary Curvature Under Zero Flux
Suppose u(0)=7, u_1=6, h=0.5, and the left boundary has zero flux. What second derivative estimate follows from the ghost-point stencil (u_{-1}-2u_0+u_1)/h^2?
打开 →题目3583 · 随机过程
Brownian Motion Divided by a Deterministic Clock
Apply It\^o to $X_t=W_t/(1+t)$. What is $dX_t$?
打开 →题目1956 · 数学
Carry Drag Versus Quadratic Inventory 11
A carry term explodes as the state approaches -1, so the desk cannot simply push x downward. Minimize H(x) = 4 x^2 + 9/(1+x) over x > -1.
打开 →题目3613 · 随机过程
Centered OU Dynamics
An OU process satisfies dX_t = 1.4(3 - X_t)dt + 0.7 dW_t. If Y_t = X_t - 3, what SDE does Y_t satisfy?
打开 →题目3391 · 数学
Central-Difference Richardson Upgrade 1
A second-order central-difference estimate of f'(x) is 1.28 at h=0.2 and 1.22 at h=0.1. Using an O(h^2) Richardson extrapolation, what improved estimate do you get?
打开 →题目3653 · 随机过程
Choose c for a Cubic Brownian Local Martingale
For standard Brownian motion, choose $c$ so that $X_t=W_t^3-c\int_0^t W_s\,ds$ is a local martingale.
打开 →题目3655 · 随机过程
Choose c in the Quartic Brownian Polynomial
For standard Brownian motion, choose $c$ so that $X_t=W_t^4-6tW_t^2+ct^2$ is a martingale.
打开 →题目3615 · 随机过程
CIR Exponential Rescaling
A CIR process satisfies dX_t = 1.2(4 - X_t)dt + 0.5 sqrt(X_t)dW_t. If Y_t = e^{1.2 t} X_t, what SDE does Y_t satisfy?
打开 →题目3258 · 数学
Classify the Origin for -x^2-4y^2
Classify the critical point at the origin for $f(x,y)=-x^2-4y^2$.
打开 →题目3260 · 数学
Classify the Origin for -x^2+2xy-y^2
Classify the critical point at the origin for $f(x,y)=-x^2+2xy-y^2$.
打开 →题目3257 · 数学
Classify the Origin for x^2-y^2
Classify the critical point at the origin for $f(x,y)=x^2-y^2$.
打开 →题目3256 · 数学
Classify the Origin for x^2+2y^2
Classify the critical point at the origin for $f(x,y)=x^2+2y^2$.
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