#P04 · Probability, Statistics & AI · Lv.2View in 3D Mind-Map

Expected Value & Variance

Christiaan Huygens & Pafnuty Chebyshev · 1657 CE

E[X] = \sum_i x_i p_i, \quad \operatorname{Var}(X) = E[(X-\mu)^2] = E[X^2] - (E[X])^2

\sigma_X = \sqrt{\operatorname{Var}(X)}, \quad E[aX+b] = aE[X]+b

Measures the center of mass (mean μ) and the squared spread around the mean (variance σ²) of a random variable.

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Payoff x₁ (p=0.5)

Payoff x₂ (p=0.3)

Payoff x₃ (p=0.2)

E[X] = 10.00, \quad \operatorname{Var}(X) = 175.0 - (10.0)^2 = 75.00

기댓값 E[X] (평균 μ)

10.000

분산 Var(X) = σ²

75.000

표준편차 σ

8.660

Prerequisite Path (7 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#P01Permutations & Combinations

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

STEP 3#A02Polynomial Expansion & Factoring

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

STEP 4#A03Laws of Exponents & Radicals

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

STEP 5#P02Conditional Probability & Chain Rule

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

STEP 6#A08Binomial Theorem & Pascal's Triangle

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

STEP 7#P04Expected Value & Variance

E[X] = \sum_i x_i p_i

🚀 Next Unlocks (Derived Formulas) (5)

#P05Binomial Distribution #P06Normal (Gaussian) Distribution & Z-Score #P07Central Limit Theorem & Confidence Intervals #P08Pearson Correlation & Linear Regression (OLS)#P09Shannon Entropy & Cross-Entropy Loss

Geometric & Intuitive Insight

'제곱의 평균 빼기 평균의 제곱(E[X²] - E[X]²)'은 매번 편차를 구하지 않고도 한 번의 순회로 분산을 빠르게 계산하게 해 줍니다.

Step-by-Step Derivation

Step 1.Var(X) = E[(X - μ)²] = E[X² - 2μX + μ²] (단, μ = E[X])
Step 2.기댓값의 선형성에 의해 E[X²] - 2μE[X] + μ² = E[X²] - 2μ² + μ²
Step 3.따라서 Var(X) = E[X²] - (E[X])²

AI & Coding Real-World Use

Core to Reinforcement Learning expected reward maximization, Batch/Layer Normalization, and Xavier/He weight initialization.

#Reinforcement Learning#Batch Normalization#Quant Risk#Weight Initialization

const mean = xs.reduce((a,b)=>a+b,0)/xs.length; const variance = xs.reduce((s,x)=>s+(x-mean)**2,0)/xs.length;

Probability, Statistics & AI — Related Formulas in This Cluster

Permutations & Combinations

Lv.1 · Combinatorics

#P01

Conditional Probability & Chain Rule

Lv.2 · LLM Next-Token

Binomial Distribution

Lv.2 · A/B Testing

#P05

Normal (Gaussian) Distribution & Z-Score

Lv.3 · Diffusion Models

#P06

Central Limit Theorem & Confidence Intervals

Lv.4 · Mini-Batch SGD

#P07