#P07 · Probability, Statistics & AI · Lv.4View in 3D Mind-Map

Central Limit Theorem & Confidence Intervals

Pierre-Simon Laplace & Aleksandr Lyapunov · 1810 CE

\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \mathcal{N}(0, 1) \quad (n \to \infty)

\text{95\% CI} = \bar{X} \pm 1.96 \frac{\sigma}{\sqrt{n}}

States that the sampling distribution of the sample mean approaches a normal distribution with standard error σ/√n as sample size n grows, regardless of the population shape.

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Sample Mean X̄

Std Dev σ

Sample Size n

\text{SE} = \frac{12}{\sqrt{36}} = 2.000, \quad \text{95\% CI} = 50 \pm 3.92

표준오차 SE = σ/√n

2.0000

95% 신뢰구간 [L, U]

[46.08, 53.92]

오차 한계 (±1.96 SE)

±3.920

Prerequisite Path (16 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#P01Permutations & Combinations

{}_nP_r = \frac{n!}{(n-r)!}

STEP 2#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 3#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 4#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 5#A06Logarithm Laws & Change of Base

\log_a(xy) = \log_a x + \log_a y

STEP 6#P02Conditional Probability & Chain Rule

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

STEP 7#A08Binomial Theorem & Pascal's Triangle

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

STEP 8#C01Limits, Continuity & Euler's Number e

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828

STEP 9#P05Binomial Distribution

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

STEP 10#A07Arithmetic & Geometric Series Sum

S_n = \frac{a(1 - r^n)}{1 - r}

STEP 11#C02Derivative Definition & Instantaneous Rate

f'(x) = \frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

STEP 12#P04Expected Value & Variance

E[X] = \sum_i x_i p_i

STEP 13#C04Fundamental Theorem of Calculus (FTC)

\int_a^b f(x)\

STEP 14#P06Normal (Gaussian) Distribution & Z-Score

f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)

STEP 15#C06Taylor & Maclaurin Series Expansion

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

STEP 16#P07Central Limit Theorem & Confidence Intervals

\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \mathcal{N}(0

Geometric & Intuitive Insight

표본 오차를 절반으로 줄이려면 표본 크기 n을 2배가 아니라 4배(√4 = 2)로 늘려야 한다는 '루트 n의 법칙'을 알려줍니다.

Step-by-Step Derivation

Step 1.독립 항등분포 확률변수 X₁..Xₙ의 표본평균 X̄ = (X₁+...+Xₙ)/n에 대해 E[X̄] = μ입니다.
Step 2.분산 성질 Var(aX) = a²Var(X)에 의해 Var(X̄) = (1/n²) · nσ² = σ²/n이므로 표준오차 SE = σ/√n입니다.
Step 3.특성함수(푸리에 변환)의 테일러 2차 전개에 의해 n→∞일 때 표준화된 합은 N(0,1)로 수렴합니다.

AI & Coding Real-World Use

Explains the 1/√n variance reduction in Mini-Batch SGD gradients and Monte Carlo Ray Tracing error bounds.

#Mini-Batch SGD#Monte Carlo Ray Tracing#Confidence Intervals#Statistical Inference

const stdError = (sigma, n) => sigma / Math.sqrt(n); const ci95 = (mean, se) => [mean - 1.96*se, mean + 1.96*se];

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Binomial Distribution

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