Pre-U CAIE Statistics: Quick Reference of Formulas and Theorems | Pre-U CAIE 统计：公式定理速查手册

📚 Pre-U CAIE Statistics: Quick Reference of Formulas and Theorems | Pre-U CAIE 统计：公式定理速查手册

This quick reference guide covers essential formulas and theorems for the Pre-U CAIE Statistics syllabus. It is designed for rapid revision and consolidation of key concepts in probability, distributions, inference, and regression. All notation follows standard conventions used in examination papers.

本速查手册涵盖 Pre-U CAIE 统计课程的核心公式与定理，旨在帮助考生快速复习和巩固概率、分布、推断和回归等关键概念。所有符号遵循考试用标准惯例。

1. Probability Basics | 概率基础

Sample space and probability axioms: For any event A, 0 ≤ P(A) ≤ 1, and P(S) = 1 where S is the sample space.

样本空间与概率公理：对任意事件 A，0 ≤ P(A) ≤ 1，且 P(S) = 1，其中 S 为样本空间。

Complement rule: P(A’) = 1 – P(A).

互补规则：P(A’) = 1 – P(A)。

Addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). For mutually exclusive events, P(A ∪ B) = P(A) + P(B).

加法公式：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若 A 与 B 互斥，则 P(A ∪ B) = P(A) + P(B)。

Conditional probability: P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

条件概率：P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。

Independence: Events A and B are independent if and only if P(A ∩ B) = P(A) P(B).

独立性：A 与 B 独立当且仅当 P(A ∩ B) = P(A) P(B)。

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2. Discrete Random Variables | 离散随机变量

Probability mass function (PMF): p(x) = P(X = x) satisfies Σ p(x) = 1 and 0 ≤ p(x) ≤ 1.

概率质量函数：p(x) = P(X = x)，满足 Σ p(x) = 1 且 0 ≤ p(x) ≤ 1。

Expected value (mean): E(X) = μ = Σ x p(x).

期望值（均值）：E(X) = μ = Σ x p(x)。

Variance: Var(X) = E[(X – μ)²] = E(X²) – [E(X)]².

方差：Var(X) = E[(X – μ)²] = E(X²) – [E(X)]²。

Linear transformations: E(aX + b) = a E(X) + b, Var(aX + b) = a² Var(X).

线性变换：E(aX + b) = a E(X) + b，Var(aX + b) = a² Var(X)。

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3. Continuous Random Variables | 连续随机变量

Probability density function (PDF): f(x) ≥ 0, and ∫_-∞^∞ f(x) dx = 1. Probability of an interval: P(a < X < b) = ∫_a^b f(x) dx.

概率密度函数：f(x) ≥ 0，且 ∫_-∞^∞ f(x) dx = 1。区间概率：P(a < X < b) = ∫_a^b f(x) dx。

Cumulative distribution function (CDF): F(x) = P(X ≤ x) = ∫_-∞^x f(t) dt.

累积分布函数：F(x) = P(X ≤ x) = ∫_-∞^x f(t) dt。

Expectation and variance: E(X) = ∫ x f(x) dx, Var(X) = ∫ (x – μ)² f(x) dx = E(X²) – μ².

期望与方差：E(X) = ∫ x f(x) dx，Var(X) = ∫ (x – μ)² f(x) dx = E(X²) – μ²。

Median and percentiles: The median m satisfies F(m) = 0.5. The 100p-th percentile is the value x such that F(x) = p.

中位数与百分位数：中位数 m 满足 F(m) = 0.5。第 100p 百分位数为满足 F(x) = p 的 x 值。

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4. Common Discrete Distributions | 常见离散分布

Binomial distribution X ~ B(n, p): P(X = k) = C(n, k) p^k (1 – p)^{n – k}, k = 0,1,…,n. E(X) = np, Var(X) = np(1 – p). Conditions: fixed number n of independent trials, constant success probability p.

二项分布 X ~ B(n, p)：P(X = k) = C(n, k) p^k (1 – p)^{n – k}, k = 0,1,…,n。E(X) = np，Var(X) = np(1 – p)。条件：固定试验次数 n，各次试验独立，每次成功概率 p 恒定。

Poisson distribution X ~ Po(λ): P(X = k) = λ^k e^-λ / k!, k = 0,1,2,… E(X) = Var(X) = λ. Used for rare events in a fixed interval.

泊松分布 X ~ Po(λ)：P(X = k) = λ^k e^-λ / k!，k = 0,1,2,…。E(X) = Var(X) = λ。适用于固定区间内稀有事件。

Geometric distribution X ~ Geo(p) (number of trials to first success): P(X = k) = (1 – p)^{k – 1} p, k = 1,2,3,… E(X) = 1/p, Var(X) = (1 – p) / p².

几何分布 X ~ Geo(p)（首次成功所需的试验次数）：P(X = k) = (1 – p)^{k – 1} p，k = 1,2,3,…。E(X) = 1/p，Var(X) = (1 – p) / p²。

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5. Common Continuous Distributions | 常见连续分布

Uniform distribution X ~ U(a, b): f(x) = 1/(b – a) for a ≤ x ≤ b. E(X) = (a + b)/2, Var(X) = (b – a)²/12.

均匀分布 X ~ U(a, b)：f(x) = 1/(b – a)，a ≤ x ≤ b。E(X) = (a + b)/2，Var(X) = (b – a)²/12。

Exponential distribution X ~ Exp(λ): f(x) = λ e^{-λ x}, x ≥ 0. E(X) = 1/λ, Var(X) = 1/λ². Memoryless property: P(X > s + t | X > s) = P(X > t).

指数分布 X ~ Exp(λ)：f(x) = λ e^{-λ x}，x ≥ 0。E(X) = 1/λ，Var(X) = 1/λ²。无记忆性：P(X > s + t | X > s) = P(X > t)。

Normal distribution X ~ N(μ, σ²): f(x) = 1/[σ √(2π)] e^{-(x – μ)²/(2σ²)}. Standard normal Z = (X – μ)/σ ~ N(0, 1). If X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²) are independent, then X + Y ~ N(μ₁ + μ₂, σ₁² + σ₂²).

正态分布 X ~ N(μ, σ²)：f(x) = 1/[σ √(2π)] e^{-(x – μ)²/(2σ²)}。标准正态 Z = (X – μ)/σ ~ N(0, 1)。若独立随机变量 X ~ N(μ₁, σ₁²) 且 Y ~ N(μ₂, σ₂²)，则 X + Y ~ N(μ₁ + μ₂, σ₁² + σ₂²)。

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6. Sampling and Central Limit Theorem | 抽样与中心极限定理

Distribution of the sample mean (normal population): If X ~ N(μ, σ²), then x̄ ~ N(μ, σ²/n) for a random sample of size n. Standard error = σ/√n.

样本均值的分布（正态总体）：若 X ~ N(μ, σ²)，则对于样本容量 n，x̄ ~ N(μ, σ²/n)。标准误 = σ/√n。

Central Limit Theorem (CLT): For a random sample of size n from any population with mean μ and variance σ², the sample mean x̄ is approximately N(μ, σ²/n) when n is large (usually n ≥ 30).

中心极限定理：对于来自均值为 μ、方差为 σ² 的任意总体的容量为 n 的随机样本，当 n 充分大时（通常 n ≥ 30），样本均值 x̄ 近似服从 N(μ, σ²/n)。

Sample proportion: If X ~ B(n, p), the sample proportion p̂ = X/n has E(p̂) = p, Var(p̂) = p(1 – p)/n. For large n (np > 5, n(1 – p) > 5), p̂ is approximately N(p, p(1 – p)/n).

样本比例：若 X ~ B(n, p)，样本比例 p̂ = X/n 满足 E(p̂) = p，Var(p̂) = p(1 – p)/n。当 n 较大时 (np > 5, n(1 – p) > 5)，p̂ 近似服从 N(p, p(1 – p)/n)。

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7. Estimation | 估计

Point estimates and unbiasedness: A statistic θ̂ is an unbiased estimator of θ if E(θ̂) = θ. For example, x̄ is unbiased for μ; s² = Σ(x – x̄)²/(n – 1) is unbiased for σ².

点估计与无偏性：若统计量 θ̂ 满足 E(θ̂) = θ，则它是 θ 的无偏估计量。例如，x̄ 是 μ 的无偏估计；s² = Σ(x – x̄)²/(n – 1) 是 σ² 的无偏估计。Published by TutorHao | Pre-U 统计 Revision Series | aleveler.com

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