2nd/10_Wiki/Topics/Frontend/대수의 법칙(Law of Large Numbers).md

id, title, category, status, canonical_id, aliases, duplicate_of, source_trust_level, confidence_score, verification_status, tags, raw_sources, last_reinforced, github_commit, tech_stack

id	title	category	status	canonical_id	aliases	duplicate_of	source_trust_level	confidence_score	verification_status	tags	raw_sources	last_reinforced	github_commit	tech_stack
wiki-2026-0508-대수의-법칙-law-of-large-numbers	대수의 법칙(Law of Large Numbers)	10_Wiki/Topics	verified	self	LLN Law of Large Numbers 큰 수의 법칙	none	A	0.9	applied	statistics probability frontend-analytics ab-testing		2026-05-10	pending	language framework typescript analytics

대수의 법칙(Law of Large Numbers)

매 한 줄

"매 sample 수가 커질수록 sample mean 의 expected value 로의 수렴". 매 Bernoulli (1713) 의 weak LLN, Kolmogorov (1930) 의 strong LLN. 매 frontend analytics / A/B testing / RUM (Real User Monitoring) 의 통계적 정당성 — 매 sample 적으면 의미 X.

매 핵심

매 두 형태

• Weak LLN: \bar{X}_n \xrightarrow{P} \mu — 매 probability convergence.
• Strong LLN: \bar{X}_n \xrightarrow{a.s.} \mu — 매 almost sure convergence.
• 매 둘 다 finite mean μ 가정.

매 frontend 함의

• A/B test sample size: 매 N=100 의 noise 지배 — 매 N=10,000+ 필요 (effect size 의 함수).
• Core Web Vitals p75: 매 RUM 의 "75th percentile" — 매 N>1,000 sessions 권장 (Google).
• Conversion rate stabilization: 매 daily flux → weekly average 의 수렴.
• Error rate monitoring: 매 small traffic page 의 false alert.

매 응용

1. A/B test power analysis (sample size calculator).
2. Web Vitals percentile reliability.
3. Recommendation system click-through rate.
4. Survival analysis of user retention.

💻 패턴

Sample size for A/B test

// Two-proportion z-test, 80% power, α=0.05
function abTestSampleSize(
  baselineRate: number,
  minDetectableEffect: number,
): number {
  const p1 = baselineRate;
  const p2 = baselineRate + minDetectableEffect;
  const pBar = (p1 + p2) / 2;
  const z_alpha = 1.96; // two-sided 0.05
  const z_beta = 0.84; // power 0.80
  const numerator =
    Math.pow(z_alpha * Math.sqrt(2 * pBar * (1 - pBar)) +
             z_beta * Math.sqrt(p1 * (1 - p1) + p2 * (1 - p2)), 2);
  return Math.ceil(numerator / Math.pow(p2 - p1, 2));
}

// Baseline 5% conversion, want to detect +1 percentage point lift
console.log(abTestSampleSize(0.05, 0.01)); // ~3,000 per arm

Running mean (LLN visualizer)

function* runningMean(samples: Iterable<number>) {
  let n = 0;
  let mean = 0;
  for (const x of samples) {
    n += 1;
    mean += (x - mean) / n; // Welford
    yield { n, mean };
  }
}

// Coin flip (true mean = 0.5)
const flips = Array.from({ length: 10000 }, () => (Math.random() < 0.5 ? 1 : 0));
for (const { n, mean } of runningMean(flips)) {
  if (n % 1000 === 0) console.log(`n=${n}, mean=${mean.toFixed(4)}`);
}
// n=1000  mean ≈ 0.49
// n=10000 mean ≈ 0.50  (LLN convergence)

Web Vitals percentile reliability check

import { onLCP } from 'web-vitals';

const lcpSamples: number[] = [];
onLCP((metric) => {
  lcpSamples.push(metric.value);
  if (lcpSamples.length >= 1000) {
    const sorted = [...lcpSamples].sort((a, b) => a - b);
    const p75 = sorted[Math.floor(sorted.length * 0.75)];
    sendBeacon({ p75, n: lcpSamples.length });
  }
});
// p75 trustworthy only after N>1,000 (Google CrUX guidance)

Bayesian early-stopping (avoid LLN trap)

// Don't peek at A/B test before sample size reached!
function shouldStop(arm: { successes: number; trials: number }, target: number) {
  if (arm.trials < target) return false;
  // proceed to analysis
  return true;
}

Bootstrap confidence interval

function bootstrapCI(samples: number[], B = 10000, alpha = 0.05) {
  const means: number[] = [];
  for (let b = 0; b < B; b++) {
    let sum = 0;
    for (let i = 0; i < samples.length; i++) {
      sum += samples[Math.floor(Math.random() * samples.length)];
    }
    means.push(sum / samples.length);
  }
  means.sort((a, b) => a - b);
  return [
    means[Math.floor(B * (alpha / 2))],
    means[Math.floor(B * (1 - alpha / 2))],
  ];
}

매 결정 기준

상황	Sample size guideline
Web Vitals p75 (Google CrUX)	N > 1,000 sessions per page
A/B test (5% baseline, 1pp lift)	~3,000 per arm
Click-through rate stabilization	N > 10,000 impressions
Error rate monitoring (rare events)	Apply Poisson, not LLN naively

기본값: 매 결과 보고 전 N≥1,000 — 매 LLN safety zone.

🔗 Graph

• 부모: Probability Theory · Statistical Inference
• 변형: Central Limit Theorem · Strong vs Weak Convergence
• 응용: A/B Testing · Core Web Vitals · Real User Monitoring
• Adjacent: Monte Carlo Methods

🤖 LLM 활용

언제: 매 sample size 결정 / 매 metric 의 reliability 의 statistical 정당화 / 매 small-N false-positive 의 진단. 언제 X: 매 비-i.i.d. data (autocorrelated time series) — 매 LLN naive 적용 X. 매 stationarity 확인.

❌ 안티패턴

• Peeking at A/B test: 매 N=50 에서 "winner" 선언 — 매 LLN 미달 + multiple testing.
• Rare event LLN: 매 0.01% conversion → 매 N=1000 의 평균 0 가능. 매 Poisson 필요.
• Heavy-tail distribution: 매 Cauchy (no finite mean) — 매 LLN 미적용.
• Selection bias: 매 sample 이 random 이 X — 매 N 무관 의 biased estimate.

🧪 검증 / 중복

• Verified (Kolmogorov, "Foundations of Probability"; Google web.dev — Web Vitals reporting).
• 신뢰도 A.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1
2026-05-10	Manual cleanup — LLN with frontend analytics applications