Year 13 CAIE Statistics: Core Knowledge Points Review | Year 13 CAIE 统计：核心知识点梳理

📚 Year 13 CAIE Statistics: Core Knowledge Points Review | Year 13 CAIE 统计：核心知识点梳理

Year 13 CAIE Statistics deepens your mastery of probability models, sampling theory and formal hypothesis testing. This article reviews the core topics of the Probability & Statistics 2 (Paper 6) syllabus, highlighting key formulas, conditions and applications to help you build a secure foundation for examination success.

Year 13 CAIE 统计深化了对概率模型、抽样理论和规范假设检验的掌握。本文围绕概率与统计 2（试卷 6）大纲，梳理核心知识点，突出关键公式、适用条件与应用，助力学生构建扎实的应考基础。

1. Poisson Distribution | 泊松分布

The Poisson distribution models the number of events, X, occurring in a fixed interval of time or space. Events must be random, independent and occur at a constant average rate λ. The probability mass function is:

泊松分布用于描述在固定时间或空间间隔内发生的事件次数 X。事件必须随机、独立且以恒定平均速率 λ 发生。其概率质量函数为：

P(X = x) = e^-λ λ^x / x! for x = 0, 1, 2, …

Key properties: E(X) = Var(X) = λ. If X_i ~ Po(λ_i) are independent, then ΣX_i ~ Po(Σλ_i). Conditions for a Poisson model also require that events occur singly – two or more events cannot happen at exactly the same instant. Typical applications include the number of phone calls per minute, flaws per metre of cloth, or accidents per week.

主要性质：E(X) = Var(X) = λ。若 X_i ~ Po(λ_i) 相互独立，则 ΣX_i ~ Po(Σλ_i)。泊松模型的条件还要求事件单独发生——同一瞬间不能同时发生两个或多个事件。典型应用包括每分钟电话呼叫次数、每米布匹瑕疵数、每周事故数等。

2. Approximations Using Poisson and Normal | 泊松与正态近似

When a binomial distribution Bin(n, p) has large n and very small p, such that np < 5 and n > 50, it can be approximated by Po(np). Because both distributions are discrete, no continuity correction is needed. For a Poisson distribution with large λ (usually λ > 15), a normal approximation N(λ, λ) is commonly used. Here a continuity correction is essential because a continuous distribution is used to approximate a discrete one.

当二项分布 Bin(n, p) 的 n 很大且 p 很小，且满足 np < 5、n > 50 时，可用 Po(np) 进行近似。由于两者均为离散分布，无需连续性修正。对于 λ 较大（通常 λ > 15）的泊松分布，常用正态分布 N(λ, λ) 近似。此时必须采用连续性修正，因为是用连续分布近似离散分布。

To apply the correction: P(X ≤ x) ≈ P(N < x + 0.5), P(X ≥ x) ≈ P(N > x − 0.5), and P(X = x) ≈ P(x − 0.5 < N < x + 0.5). These adjustments greatly improve accuracy, especially for tail probabilities.

应用修正时：P(X ≤ x) ≈ P(N < x + 0.5)，P(X ≥ x) ≈ P(N > x − 0.5)，P(X = x) ≈ P(x − 0.5 < N < x + 0.5)。这些调整能显著提高精度，尤其是在尾部概率中。

3. Continuous Random Variables: PDF and CDF | 连续型随机变量：概率密度函数与累积分布函数

A continuous random variable X is described by its probability density function f(x), where f(x) ≥ 0 and the total area under f(x) equals 1: ∫_-∞^∞ f(x) dx = 1. Probabilities are represented by areas under the density curve: P(a < X < b) = ∫_a^b f(x) dx.

连续型随机变量 X 由其概率密度函数 f(x) 描述，f(x) ≥ 0 且曲线下的总面积为

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