Year 13 CIE Statistics: Formula & Theorem Quick Reference Handbook

📚 Year 13 CIE Statistics: Formula & Theorem Quick Reference Handbook | 公式定理速查手册

This compact handbook gathers the essential formulas and theorems for Year 13 CIE A-Level Statistics (9709). It is designed as a quick reference for revision, covering probability generating functions, continuous distributions, sampling theory, confidence intervals, hypothesis tests and chi-squared tests. Each formula is paired with a concise explanation in both English and Chinese, strictly following CIE syllabus conventions.

本速查手册汇集了 Year 13 CIE A-Level 统计学 (9709) 的核心公式与定理，专为高效复习而设计。内容覆盖概率生成函数、连续分布、抽样理论、置信区间、假设检验以及卡方检验。每个公式都配有中英双语简明解释，严格遵循 CIE 考纲要求。

1. Probability Generating Functions (PGFs) | 概率生成函数

The probability generating function of a non-negative integer-valued discrete random variable X is G_X(t) = E(t^X) = ∑_x t^x P(X = x), |t| ≤ 1.

非负整数值离散随机变量 X 的概率生成函数为 G_X(t) = E(t^X) = ∑_x t^x P(X = x)，其中 |t| ≤ 1。

E(X) = G’_X(1)

The first derivative evaluated at t=1 gives the expectation. This is a fundamental PGF property for computing the mean.

一阶导数在 t=1 处的值即为期望值。这是利用 PGF 计算均值的基本性质。

Var(X) = G”_X(1) + G’_X(1) − [G’_X(1)]²

The variance is obtained from the first two derivatives of the PGF at t=1. Note that this formula is specific to PGFs.

方差由 PGF 在 t=1 处的前两阶导数组合得到。注意该公式专用于概率生成函数。

If X and Y are independent, then G_X+Y(t) = G_X(t) × G_Y(t). This multiplicative property simplifies handling sums of independent variables.

若 X 和 Y 独立，则 G_X+Y(t) = G_X(t) × G_Y(t)。这一乘法性质大大简化了独立变量和的处理。

PGF of Binomial(n,p): (q + pt)ⁿ | 二项分布的 PGF: (q + pt)ⁿ
PGF of Poisson(λ): e^λ(t−1) | 泊松分布的 PGF: e^λ(t−1)

2. Geometric & Negative Binomial Distributions | 几何分布与负二项分布

The geometric distribution models the number of trials up to (and including) the first success. P(X = x) = p q^x−1, x = 1,2,3… where q = 1 − p.

几何分布模拟的是直到（包括）首次成功所需的试验次数。P(X = x) = p q^x−1，x = 1,2,3…，其中 q = 1 − p。

E(X) = 1/p, Var(X) = q/p²

The PGF for the geometric distribution is G_X(t) = pt / (1 − qt). This is derived by summing the geometric series.

几何分布的 PGF 为 G_X(t) = pt / (1 − qt)，可通过等比数列求和推导得出。

The negative binomial distribution counts the number of trials needed to obtain exactly r successes. P(X = x) = ^x−1C_r−1 p^r q^x−r, x = r, r+1, …

负二项分布统计的是获得 r 次成功所需的总试验次数。P(X = x) = ^x−1C_r−1 p^r q^x−r，x = r, r+1, …

E(X) = r/p, Var(X) = rq/p²

The PGF of a negative binomial distribution is [pt/(1 − qt)]^r, which follows directly from the PGF of a single geometric variable raised to the power r.

负二项分布的 PGF 为 [pt/(1 − qt)]^r，这相当于单个几何分布的 PGF 的 r 次幂。

3. Continuous Random Variables: PDF | 连续随机变量：概率密度函数

A continuous random variable X has probability density function f(x) such that f(x) ≥ 0 and the total area under the curve equals 1.

连续随机变量 X 具有概率密度函数 f(x)，满足 f(x) ≥ 0 且曲线下的总面积为 1。

∫_−∞^∞ f(x) dx = 1

For any interval, P(a < X < b) = ∫_a^b f(x) dx. Always check that the cumulative probability sums to 1 when verifying a PDF.

对于任意区间，P(a < X < b) = ∫_a^b f(x) dx。验证 PDF 时务必确认累积概率之和为 1。

The probability at a single point is zero for a continuous distribution: P(X = c) = 0.

对于连续分布，单点概率恒为零：P(X = c) = 0。

Linear PDF on [a,b]: f(x) = 1/(b − a) | 区间 [a,b] 上的均匀分布 PDF: f(x) = 1/(b − a)
f(x) may be defined piecewise across intervals | f(x) 可以在不同区间分段定义

4. Cumulative Distribution Function (CDF) | 累积分布函数

The cumulative distribution function is F(x) = P(X ≤ x) = ∫_−∞^x f(t) dt. F(−∞) = 0 and F(∞) = 1.

累积分布函数为 F(x) = P(X ≤ x) = ∫_−∞^x f(t) dt。F(−∞) = 0，F(∞) = 1。

For any a < b, P(a < X ≤ b) = F(b) − F(a). The derivative of the CDF gives back the PDF: f(x) = F'(x).

对于任意 a < b，P(a < X ≤ b) = F(b) − F(a)。CDF 的导数即为 PDF：f(x) = F'(x)。

The median m satisfies F(m) = 0.5. The upper quartile Q₃ and lower quartile Q₁ satisfy F(Q₃) = 0.75 and F(Q₁) = 0.25 respectively.

中位数 m 满足 F(m) = 0.5。上四分位数 Q₃ 与下四分位数 Q₁ 分别满足 F(Q₃) = 0.75 和 F(Q₁) = 0.25。

When solving for percentiles, always find the inverse of the CDF while respecting the piecewise definitions.

求解百分位数时，需根据分段函数的定义求 CDF 的反函数。

5. Expectation and Variance | 期望与方差

For a continuous variable: E(X) = ∫_−∞^∞ x f(x) dx and E(g(X)) = ∫_−∞^∞ g(x) f(x) dx.

对于连续变量：E(X) = ∫_−∞^∞ x f(x) dx，E(g(X)) = ∫_−∞^∞ g(x) f(x) dx。

Var(X) = E(X²) − [E(X)]²

This short-cut formula for variance holds for both discrete and continuous distributions. Always compute E(X²) by integration or summation.

方差的简化计算公式同时适用于离散和连续分布，需通过积分或求和计算 E(X²)。

For a linear transformation Y = aX + b: E(Y) = a E(X) + b, and Var(Y) = a² Var(X).

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

Standard deviation σ = √Var(X). The units of variance are the square of the units of X, so reporting standard deviation is often more interpretable.

标准差 σ = √Var(X)。方差的单位是 X 单位的平方，因此报告标准差通常更易于解释。

6. Normal Distribution | 正态分布

If X ~ N(μ, σ²), its probability density function is f(x) = (1/(σ√(2π))) e^{−(x−μ)²/(2σ²)}.

若 X ~ N(μ, σ²)，其概率密度函数为 f(x) = (1/(σ√(2π))) e^{−(x−μ)²/(2σ²)}。

Z = (X − μ) / σ ~ N(0,1)

Standardisation converts any normal variable to the standard normal distribution. Use the Φ(z) table to find probabilities.

标准化可将任意正态变量转换为标准正态分布，然后使用 Φ(z) 表查找概率。

Symmetric intervals: P(−a < Z < a) = 2Φ(a) − 1. The sum of independent normal variables is also normal: X±Y ~ N(μ_X±μ_Y, σ_X²+σ_Y²).

对称区间：P(−a < Z < a) = 2Φ(a) − 1。独立正态变量之和仍服从正态分布：X±Y ~ N(μ_X±μ_Y, σ_X²+σ_Y²)。

7. Exponential Distribution | 指数分布

An exponential random variable X with rate parameter λ has PDF f(x) = λ e^{−λ x} for x ≥ 0, and 0 otherwise.

速率为 λ 的指数分布随机变量 X 的概率密度函数为 f(x) = λ e^{−λ x}，x ≥ 0；其他情况为零。

E(X) = 1/λ, Var(X) = 1/λ²

The cumulative distribution function is F(x) = 1 − e^{−λ x}, x ≥ 0. The exponential distribution is memoryless: P(X > s+t | X > s) = P(X > t).

累积分布函数为 F(x) = 1 − e^{−λ x}，x ≥ 0。指数分布具有无记忆性：P(X > s+t | X > s) = P(X > t)。

Median m = (ln 2)/λ ≈ 0.693/λ. This distribution is often used for waiting times.

中位数 m = (ln 2)/λ ≈ 0.693/λ。该分布常用于模拟等待时间。

8. Sampling Distributions & CLT | 抽样分布与中心极限定理

For a random sample of size n from a population with mean μ and variance σ², the sample mean X̄ satisfies E(X̄) = μ and Var(X̄) = σ²/n.

对于来自均值为 μ、方差为 σ² 的总体、容量为 n 的随机样本，样本均值 X̄ 满足 E(X̄) = μ，Var(X̄) = σ²/n。

If the population is normal, then X̄ ~ N(μ, σ²/n) exactly for any n.

若总体服从正态分布，则对于任何 n，均有 X̄ ~ N(μ, σ²/n)。

Central Limit Theorem: For large n, X̄ ≈ N(μ, σ²/n)

The Central Limit Theorem states that the sampling distribution of the sample mean approximates a normal distribution as n increases, usually n ≥ 30.

中心极限定理指出：随着样本容量增大，样本均值的抽样分布近似服从正态分布，通常 n ≥ 30 即可近似。

Sample proportion P̂ has E(P̂) = p and Var(P̂) = p(1−p)/n. Approximate normality holds when np > 5 and n(1−p) > 5.

样本比例 P̂ 满足 E(P̂) = p，Var(P̂) = p(1−p)/n。当 np > 5 且 n(1−p) > 5 时，近似正态成立。

9. Confidence Intervals | 置信区间

Confidence intervals provide a range of plausible values for an unknown population parameter. The general form is point estimate ± critical value × standard error.

置信区间给出了未知总体参数的合理取值范围，其一般形式为：点估计 ± 临界值 × 标准误。

Mean (σ known): X̄ ± z_α/2 × σ/√n

Use this when the population variance is known and the z-value corresponds to the desired confidence level (e.g. 1.96 for 95%).

当总体方差已知时使用，z 值取决于置信水平（如 95% 置信水平下 z_0.025 = 1.96）。

Mean (σ unknown): X̄ ± t_n−1,α/2 × s/√n

When σ is estimated by sample standard deviation s, use the t-distribution with n−1 degrees of freedom.

当 σ 用样本标准差 s 估计时，必须使用自由度为 n−1 的 t 分布。

Proportion: p̂ ± z_α/2 × √(p̂(1−p̂)/n)

Large-sample confidence interval for a population proportion, provided success/failure conditions are met.

大样本下总体比例的置信区间，需满足成功/失败条件。

Poisson rate: λ̂ ± z_α/2 × √(λ̂/n)

For a Poisson process, the rate λ is estimated by the sample mean count, and the interval uses the square root of the estimated rate.

对于泊松过程，速率 λ 由样本均值计数估计，其置信区间使用该估计值的平方根。

10. Hypothesis Testing: Mean of Normal | 假设检验：正态均值

A hypothesis test assesses evidence against a null hypothesis H₀. The test statistic is compared with a critical value or used to find the p-value.

假设检验用于评估反对原假设 H₀ 的证据，通过检验统计量与临界值比较或计算 p 值进行判定。

Z-test (σ known): Z = (X̄ − μ₀) / (σ/√n)

Reject H₀: μ = μ₀ at significance level α if |Z| > z_α/2 (two-tailed) or if Z > z_α (one

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Year 13 CIE Statistics: Formula & Theorem Quick Reference Handbook | 公式定理速查手册