Pre-U CIE Statistics: Formula & Theorem Quick Reference | Pre-U CIE 统计：公式定理速查手册

📚 Pre-U CIE Statistics: Formula & Theorem Quick Reference | Pre-U CIE 统计：公式定理速查手册

This concise handbook brings together the essential formulas, distributions and inference procedures required for the CIE Pre-U Statistics course. Each section presents the core results with a clear pairing of English and Chinese explanations, making it ideal for quick review before exams.

这本速查手册汇集了 CIE Pre‑U 统计学课程的核心公式、分布与推断方法。每个部分都以英文与中文对照的方式呈现关键结论，适合考前快速回顾。

1. Basic Probability Rules | 基础概率法则

For any event A, the probability P(A) satisfies 0 ≤ P(A) ≤ 1. The universal event S has P(S) = 1.

对于任意事件 A，概率 P(A) 满足 0 ≤ P(A) ≤ 1。必然事件 S 有 P(S) = 1。

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

加法法则： P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。若 A 与 B 互斥，则 P(A ∩ B) = 0，从而 P(A ∪ B) = P(A) + P(B)。

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

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

Events A and B are independent if and only if P(A ∩ B) = P(A) P(B), equivalently P(A | B) = P(A).

事件 A 与 B 相互独立当且仅当 P(A ∩ B) = P(A) P(B)，等价地 P(A | B) = P(A)。

Law of total probability: If {B•₁, B•₂, …, B•_k} is a partition of S, then P(A) = ∑ P(A | B_i) P(B_i).

全概率公式： 若 {B•₁, B•₂, …, B_k} 是样本空间 S 的一个划分，则 P(A) = ∑ P(A | B_i) P(B_i)。

Bayes’ theorem: P(B_i | A) = P(A | B_i) P(B_i) / [∑_j P(A | B_j) P(B_j)].

贝叶斯定理： P(B_i | A) = P(A | B_i) P(B_i) / [∑_j P(A | B_j) P(B_j)]。

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2. Discrete Random Variables | 离散随机变量

A discrete random variable X has a probability mass function (PMF) p(x) = P(X = x). The cumulative distribution function is F(x) = P(X ≤ x).

离散随机变量 X 具有概率质量函数 p(x) = P(X = x)。累积分布函数为 F(x) = P(X ≤ x)。

E(X) = μ = ∑ x p(x)

期望（均值）：μ = E(X) = ∑ x p(x)。

Var(X) = σ² = E[(X − μ)²] = ∑ (x − μ)² p(x) = E(X²) − [E(X)]²

方差：Var(X) = σ² = E[(X − μ)²] = ∑ (x − μ)² p(x) = E(X²) − [E(X)]²。

The standard deviation is σ = √Var(X). For a linear transformation Y = aX + b,

E(aX + b) = a E(X) + b, Var(aX + b) = a² Var(X)

标准差为 σ = √Var(X)。线性变换 Y = aX + b 时，E(aX + b) = a E(X) + b，Var(aX + b) = a² Var(X)。

For any function g(X), E[g(X)] = ∑ g(x) p(x).

对于任意函数 g(X)，E[g(X)] = ∑ g(x) p(x)。

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3. Special Discrete Distributions | 常见离散分布

Binomial distribution X ~ B(n, p). PMF: P(X = k) = C(n, k) p^k (1 − p)^n−k, k = 0, 1, …, n.

二项分布 X ~ B(n, p)。概率质量函数：P(X = k) = C(n, k) p^k (1 − p)^n−k，k = 0, 1, …, n。

E(X) = n p, Var(X) = n p (1 − p)

期望与方差：E(X) = n p，Var(X) = n p (1 − p)。

Poisson distribution X ~ Po(λ). PMF: P(X = k) = e^−λ λ^k / k! , k = 0, 1, 2, …

泊松分布 X ~ Po(λ)。概率质量函数：P(X = k) = e^−λ λ^k / k! ，k = 0, 1, 2, …

E(X) = λ, Var(X) = λ

期望与方差均为 λ。当 n 大 p 小且 np ≈ λ 时，二项分布可用泊松分布近似。

Geometric distribution X ~ Geo(p) (number of trials to first success). PMF: P(X = k) = p (1 − p)^k−1, k = 1, 2, …

几何分布 X ~ Geo(p)（首次成功所需的试验次数）。P(X = k) = p (1 − p)^k−1，k = 1, 2, …

E(X) = 1 / p, Var(X) = (1 − p) / p²

期望：E(X) = 1/p，方差：Var(X) = (1 − p)/p²。

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4. Continuous Random Variables | 连续随机变量

A continuous random variable X is described by a probability density function (PDF) f(x), where f(x) ≥ 0 and ∫_−∞^∞ f(x) dx = 1.

连续随机变量 X 由概率密度函数 f(x) 描述，其中 f(x) ≥ 0 且 ∫_−∞^∞ f(x) dx = 1。

The cumulative distribution function (CDF) is F(x) = P(X ≤ x) = ∫_−∞^x f(t) dt. Probability of an interval: P(a < X < b) = ∫_a^b f(x) dx = F(b) − F(a).

累积分布函数 F(x) = P(X ≤ x) = ∫_−∞^x f(t) dt。区间概率：P(a < X < b) = ∫_a^b f(x) dx = F(b) − F(a)。

E(X) = μ = ∫_−∞^∞ x f(x) dx

Var(X) = σ² = ∫_−∞^∞ (x − μ)² f(x) dx = E(X²) − μ²

期望：μ = E(X) = ∫ x f(x) dx。方差：Var(X) = ∫ (x − μ)² f(x) dx = E(X²) − μ²。

The median m satisfies F(m) = 0.5; the mode is the value of x that maximises f(x).

中位数 m 满足 F(m) = 0.5；众数是使 f(x) 最大的 x 值。

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5. Normal Distribution | 正态分布

X ~ N(μ, σ²). PDF: f(x) = (1/σ√2π) exp[ −(x−μ)² / (2σ²) ].

X ~ N(μ, σ²)。概率密度函数：f(x) = (1/σ√2π) exp[ −(x−μ)² / (2σ²) ]。

Standardisation: Z = (X − μ) / σ ~ N(0, 1). The standard normal table gives Φ(z) = P(Z ≤ z).

标准化：Z = (X − μ) / σ ~ N(0, 1)。标准正态表给出 Φ(z) = P(Z ≤ z)。

Probabilities: P(X < a) = Φ((a−μ)/σ), P(a < X < b) = Φ((b−μ)/σ) − Φ((a−μ)/σ). For inverse calculations, find z such that Φ(z) = p, then x = μ + zσ.

概率计算：P(X < a) = Φ((a−μ)/σ)，P(a < X < b) = Φ((b−μ)/σ) − Φ((a−μ)/σ)。逆运算：找到 z 使 Φ(z) = p，则 x = μ + zσ。

68–95–99.7 rule: Approx. 68% within μ ± σ, 95% within μ ± 2σ, 99.7% within μ ± 3σ.

68–95–99.7 法则： 约 68% 落在 μ ± σ，约 95% 落在 μ ± 2σ，约 99.7% 落在 μ ± 3σ。

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

Consider a random sample of size n from a population with mean μ and variance σ². The sample mean is X̄ = (1/n)∑ X_i.

从均值为 μ、方差为 σ² 的总体中抽取样本量 n 的随机样本，样本均值为 X̄ = (1/n) ∑ X_i。

E(X̄) = μ, Var(X̄) = σ² / n

期望：E(X̄) = μ；方差：Var(X̄) = σ² / n。

If X ~ N(μ, σ²), then X̄ ~ N(μ, σ²/n).

若总体服从正态分布，则 X̄ ~ N(μ, σ²/n)。

Central Limit Theorem (CLT): For large n (usually n ≥ 30), X̄ is approximately N(μ, σ²/n), regardless of the population shape, provided the samples are independent and identically distributed.

中心极限定理： 当 n 足够大（通常 n ≥ 30），无论总体分布形状如何，X̄ 近似服从 N(μ, σ²/n)，前提是独立同分布样本。

Sample proportion p̂ = X/n from a binomial population: E(p̂) = p, Var(p̂) = p(1−p)/n. For large n, p̂ ∼ N(p, p(1−p)/n) approximately.

来自二项总体的样本比例 p̂ = X/n：E(p̂) = p，Var(p̂) = p(1−p)/n。n 大时，p̂ 近似服从 N(p, p(1−p)/n)。

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7. Point Estimation and Confidence Intervals | 点估计与置信区间

The sample mean x̄ is an unbiased estimator of μ. The unbiased sample variance is s² = (1/(n−1)) ∑ (x_i − x̄)².

样本均值 x̄ 是 μ 的无偏估计量。无偏样本方差为 s² = (1/(n−1)) ∑ (x_i − x̄)²。

Confidence interval for μ (σ known): x̄ ± z* ( σ / √n ), where z* is the critical value (e.g. 1.96 for 95% CI).

μ 的置信区间（σ 已知）： x̄ ± z* ( σ / √n )，其中 z* 为临界值（如 95% CI 取 1.96）。

Confidence interval for μ (σ unknown): x̄ ± t*_n−1 ( s / √n ), where t* is the critical value from the t-distribution with n−1 degrees of freedom.

μ 的置信区间（σ 未知）： x̄ ± t*_n−1 ( s / √n )，其中 t* 来自自由度为 n−1 的 t 分布。

Confidence interval for a proportion p: p̂ ± z* √( p̂(1−p̂) / n ), provided n is large and p not near 0 or 1.

比例 p 的置信区间： p̂ ± z* √( p̂(1−p̂) / n )，要求 n 大且 p 不过于接近 0 或 1。

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8. Hypothesis Testing | 假设检验

A hypothesis test assesses evidence against a null hypothesis H•₀ in favour of an alternative H•₁. The test statistic is computed from the sample, and its distribution under H•₀ determines the p-value or rejection region.

假设检验通过样本数据评估反对原假设 H•₀、支持备择假设 H•₁ 的证据。检验统计量由样本算出，其在 H•₀ 下的分布决定 p 值或拒绝域。

One-sample z-test for a mean (σ known):

单样本 z 检验（σ 已知）：

z = (x̄ − μ•₀) / ( σ / √n )

检验统计量 z = (x̄ − μ•₀) / ( σ / √n )。

One-sample t-test (σ unknown):

单样本 t 检验（σ 未知）：

t = (x̄ − μ•₀) / ( s / √n ), df = n − 1

t = (x̄ − μ•₀) / ( s / √n )，自由度 = n − 1。

Two-sample t-test (independent samples, equal variances assumed): pooled variance s_p² = [(n•₁−1)s•₁² + (n•₂−1)s•₂²] / (n•₁ + n•₂ − 2), t = (x̄•₁ − x̄•₂) / ( s_p √(1/n•₁ + 1/n•₂) ), df = n•₁ + n•₂ − 2.

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