#P06 · Probability, Statistics & AI · Lv.3View in 3D Mind-Map

Normal (Gaussian) Distribution & Z-Score

Carl Friedrich Gauss & Abraham de Moivre · 1809 CE

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

\int_{-\infty}^{\infty} f(x)\,dx = 1, \quad P(|Z|\le 1.96) \approx 0.95

The iconic bell-shaped continuous probability density governed by mean μ and standard deviation σ, ubiquitous across nature, statistics, and generative AI.

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Observed value x

Mean μ

Std Dev σ

Z = \frac{115 - 100}{15} = 1.000, \quad P(X \le 115) = 84.13\%

표준화 점수 Z-Score

+1.000σ

누적확률 P(X ≤ x)

84.13%

상위 백분위 Top %

15.87%

Prerequisite Path (14 STEP)

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

STEP 1#A01Linear Equation & Balance Axiom

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

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#P01Permutations & Combinations

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

STEP 5#A06Logarithm Laws & Change of Base

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

STEP 6#A08Binomial Theorem & Pascal's Triangle

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

STEP 7#A07Arithmetic & Geometric Series Sum

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

STEP 8#C02Derivative Definition & Instantaneous Rate

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

STEP 9#P02Conditional Probability & Chain Rule

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

STEP 10#C01Limits, Continuity & Euler's Number e

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

STEP 11#P04Expected Value & Variance

E[X] = \sum_i x_i p_i

STEP 12#P05Binomial Distribution

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

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)

🚀 Next Unlocks (Derived Formulas) (1)

#P07Central Limit Theorem & Confidence Intervals

Geometric & Intuitive Insight

평균에서 멀어질수록 확률밀도가 e^(-z²/2)의 속도로 급격히 감소하며, ±1σ 안에 약 68.3%, ±2σ 안에 약 95.4%의 데이터가 들어옵니다.

Step-by-Step Derivation

Step 1.가우스 적분 ∫_{-∞}^{∞} e^(-t²/2) dt = √(2π)를 정규화 상수로 사용해 전체 면적을 1로 맞춥니다.
Step 2.표준화 변수 Z = (X - μ)/σ로 치환하면 어떤 정규분포든 평균 0, 분산 1인 표준정규분포 N(0,1)로 변환됩니다.

AI & Coding Real-World Use

Powers Diffusion Models (Stable Diffusion) Gaussian noise scheduling, Variational Autoencoders (VAE), and 3D Gaussian Splatting (3DGS).

#Diffusion Models#3D Gaussian Splatting#VAE Latent Space#Anomaly Detection

const normalPDF = (x, mu, sig) => Math.exp(-0.5 * ((x - mu) / sig) ** 2) / (sig * Math.sqrt(2 * Math.PI));

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