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?
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中文题目题目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059 · 概率
All Triples Independent but Quadruple Not
Let $\Omega = \{0, 1, 2, \ldots, 7\}$ with uniform probability $P(\{\omega\}) = 1/8$. Write each $\omega$ in binary as $(b_2, b_1, b_0)$. Define events: $$A = \{\omega : b_0 = 1\}, \quad B = \{\omega : b_1 = 1\}, \quad C = \{\omega : b_2 = 1\}, \quad D = \{\omega : b_0 \oplus b_1
打开 →题目069 · 概率
An Event Independent of Itself Must Be Trivial
Let $A$ be an event in a probability space. (a) Write the independence condition $P(A \cap A) = P(A) \cdot P(A)$ and deduce which values of $P(A)$ satisfy it. (b) On $\Omega = \{1,2,3,4\}$ with uniform probability, verify your answer by testing $A = \{1\}$, $A = \{1,2\}$, $A = \e
打开 →题目2421 · 机器学习
Bayes Act Under Weighted Squared Loss 1
Suppose the loss is L(a,Y)=W(Y-a)^2 where W>0 is observed at prediction time. What predictor minimizes E[L(a,Y)|X,W]?
打开 →题目081 · 概率
Bertrand's Box Paradox
Three boxes each contain two coins. Box 1 has two gold coins, Box 2 has one gold and one silver coin, and Box 3 has two silver coins. You pick a box uniformly at random, then draw one coin at random from that box. The coin you draw is gold. What is the probability that the other
打开 →题目097 · 概率
Bertrand's Paradox: The Random Chord Problem
Consider a circle of radius $r$ with an inscribed equilateral triangle. A chord is drawn 'at random.' What is the probability that the chord is longer than the side of the triangle? Compute the answer under each of the following three methods of selecting a random chord: (a) **R
打开 →题目2398 · 机器学习
Bias Budget Implied by a Variance Reduction
A regularization change reduces a model's variance term from 0.30 to 0.11 while leaving irreducible noise unchanged. How much extra bias squared could you add before the total MSE stops improving?
打开 →题目090 · 概率
Borel's Paradox: Conditioning on Measure-Zero Events
Let $(\Theta, \Phi)$ be uniformly distributed on the unit sphere $S^2$, where $\Theta \in [0, 2\pi)$ is the longitude and $\Phi \in [0, \pi]$ is the colatitude, with joint density $f(\theta, \phi) = \frac{1}{4\pi} \sin \phi$. (a) Compute the conditional distribution of $\Phi$ gi
打开 →题目003 · 概率
Checkpoint-Exclusive Grid Routes
A robot moves from (0,0) to (5,3) using 5 right steps and 3 up steps in random order. How many paths visit exactly one of the checkpoints A=(2,1) and B=(4,2)?
打开 →题目2406 · 机器学习
Choose the Better Model at a Given Sample Size
At sample size n=60, compare model A with excess error 0.04 + 12/n to model B with excess error 0.16 + 2/n. Which one has smaller excess test error?
打开 →题目2458 · 机器学习
Choosing Early Stopping by the Test Curve
A team trains one model, plots test loss by boosting round, and reports the round with the best test value. Why is the final test score no longer a valid final check?
打开 →题目025 · 概率
Collision Pattern in Two-Round Sampling
In round 1, you draw 3 numbers independently and uniformly at random (with replacement) from $\{1, 2, 3, 4\}$. In round 2, you draw 2 numbers independently and uniformly at random (with replacement) from $\{3, 4, 5, 6\}$. The two pools overlap at $\{3, 4\}$. Across all 5 draws, f
打开 →题目018 · 概率
Colorings With Profile 3-2-1-0 and Different Ends
Six lockers are colored using the palette {R, G, B, Y}. Count the colorings in which one color appears 3 times, one appears 2 times, one appears once, one does not appear at all, and the first and last lockers have different colors.
打开 →题目061 · 概率
Complements of Independent Events Are Independent
Let $A$ and $B$ be independent events. Prove that $A^c$ and $B^c$ are also independent, i.e., $P(A^c \cap B^c) = P(A^c) \cdot P(B^c)$.
打开 →题目054 · 概率
Conditional Independence Breaks Under Marginalization
A coin is chosen at random: with probability $1/2$ it is fair ($p = 1/2$) and with probability $1/2$ it is biased ($p = 1$, always heads). Let $A$ be the event that the first flip is heads and $B$ the event that the second flip is heads. Show that $A$ and $B$ are conditionally in
打开 →题目026 · 概率
Conflicting Two-Screen Update
A loan is distressed with prior probability 0.3. Screen A flags a distressed loan with probability 0.8 and flags a healthy loan with probability 0.1. Any flagged loan then goes to screen B, which passes a distressed loan with probability 0.25 and passes a healthy loan with probab
打开 →题目2424 · 机器学习
Convexity of the Log-Cosh Loss 4
Show that ell(r)=ln cosh(r) is convex in the residual r.
打开 →题目2404 · 机器学习
Data Multiplier Needed to Push Variance Below a Noise Floor Fraction
A model's variance term is currently 0.30, and irreducible noise is 0.05. If variance scales exactly like 1/n, by what factor must the dataset grow so the variance term falls to 0.05?
打开 →题目2427 · 机器学习
Decision Threshold Under Asymmetric Classification Cost
A false negative costs 5 and a false positive costs 1. If p is the predicted probability of the positive class, above what threshold should you classify as positive?
打开 →题目019 · 概率
Distinct Ordered Triples With a Modular Sum and a Large Entry
How many ordered triples (a,b,c) of distinct elements chosen from {0,1,2,3,4,5,6,7,8} satisfy a+b+c ≡ 1 (mod 3) and max(a,b,c) > 5?
打开 →题目016 · 概率
Distinct Symbols in a Palindromic Access Code
A 5-character access code is formed by choosing each character independently and uniformly from $\{A, B, C, D, E, F\}$ (repetition allowed). The code is then accepted only if it is a palindrome (reads the same forwards and backwards). Among all palindromic codes, find $P(\text{at
打开 →题目022 · 概率
Divisibility Duel with Split Draws
In a two-stage experiment, you first draw 2 numbers without replacement from $\{1, 2, 3, 4, 5, 6\}$, then independently draw 1 number uniformly from $\{1, 2, 3, 4, 5\}$. Let $P$ be the product of all three drawn numbers. Find $P(6 \mid P)$, the probability that the product is div
打开 →题目073 · 概率
Divisibility Events and the Containment Trap
Let $\Omega = \{0, 1, 2, \ldots, 11\}$ with uniform probability $P(\{k\}) = 1/12$ for each $k$. Define three events based on divisibility: $A = \{k \in \Omega : 2 \mid k\}$ (even numbers), $B = \{k \in \Omega : 3 \mid k\}$ (multiples of $3$), $C = \{k \in \Omega : 4 \mid k\}$ (mu
打开 →题目039 · 概率
Double Alert Posterior
A transaction is malicious with prior probability 1/40. Two independent alert engines each fire with probability 7/10 on malicious transactions and 1/20 on benign ones. If both engines fire, what is the posterior malicious probability?
打开 →题目043 · 概率
Factory Defect Tracing and Predictive Inference
A factory has three production lines with the following output shares and defect rates: | Line | Share of output | Defect rate | |------|----------------|-------------| | 1 | 50% | 2% | | 2 | 30% | 3% | | 3 | 20% | 5% | An item is selected at random from today's output and foun
打开 →题目028 · 概率
Factory Posterior After One Good and One Bad Item
A part comes from factory A with prior probability 0.6 and from factory B otherwise. Factory A produces defects with probability 0.1; factory B produces defects with probability 0.4. Two parts from the same unknown factory are inspected and exactly one is defective. What is the p
打开 →题目2454 · 机器学习
Feature Screening Before the Split
A team ranks 5,000 candidate features by correlation with the target on the full dataset, keeps the top 30, and only then creates train and test. Why is the later split not enough to rescue the experiment?
打开 →题目075 · 概率
Fixed Points of a Random Permutation Are Not Independent
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 →题目031 · 概率
Flagged but Then Cleared
A borrower is high risk with prior probability 0.2. A fast model flags a high-risk borrower with probability 0.7 and flags a low-risk borrower with probability 0.1. Only flagged borrowers get a manual review. A high-risk flagged borrower passes manual review with probability 0.4,
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