📚 A-Level Maths Year 2 Stats and Mechanics: Key Concepts Explained | A-Level数学二年级统计与力学知识点精讲
Year 2 of A-Level Mathematics brings together advanced statistical methods and mechanics principles that are essential for further study. From probability distributions and hypothesis testing to kinematics with calculus and moments, mastering these topics is key to success. This guide breaks down each concept with clear English-Chinese explanations, helping you build a strong revision foundation.
A-Level数学二年级汇集了对于深造至关重要的高级统计方法和力学原理。从概率分布、假设检验到微积分运动学和力矩，掌握这些主题是成功的关键。本指南通过清晰的中英双语解释分解每个概念，助你打下扎实的复习基础。
1. Conditional Probability & Bayes' Theorem | 条件概率与贝叶斯定理
of an event occurring given that another event has already happened. It is defined by P(A|B) = P(A ∩ B) / P(B). Bayes' Theorem then links P(A|B) and P(B|A): P(A|B) = P(B|A)P(A) / P(B). This is used to update probabilities when new evidence arises.
条件概率是指已知另一事件已发生的情况下某事件发生的概率。其定义为 P(A|B) = P(A ∩ B) / P(B)。贝叶斯定理关联 P(A|B) 与 P(B|A)：P(A|B) = P(B|A)P(A) / P(B)。它用于在有新证据时更新概率。
s to visualise conditional probabilities. Always check whether events are independent (P(A∩B)=P(A)P(B)) or mutually exclusive (P(A∩B)=0).
在考试题中，你经常使用树状图或文氏图来可视化条件概率。务必检查事件是否独立（P(A∩B)=P(A)P(B)）或互斥（P(A∩B)=0）。
2. Discrete Probability Distributions (Binomial & Poisson) | 离散概率分布（二项与泊松）
els the number of successes in n independent trials, each with success probability p. If X ~ B(n, p), then P(X = r) = C(n, r) pr (1−p)n−r, with mean μ = np and variance σ² = np(1−p).
二项分布模拟 n 次独立试验中成功的次数，每次成功概率为 p。如果 X ~ B(n, p)，则 P(X = r) = C(n, r) pr (1−p)n−r，均值 μ = np，方差 σ² = np(1−p)。
events occurring in a fixed interval of time or space, given a constant mean rate λ. If X ~ Po(λ), then P(X = r) = e−λ λr / r! is often used as an approximation to the binomial when n is large and p is small (np ≈ λ).
泊松分布模拟在固定时间或空间间隔内事件发生的次数，给定恒定平均率 λ。如果 X ~ Po(λ)，则 P(X = r) = e−λ λr / r!，均值=方差=λ。它常被用作二项分布的近似，当 n 大且 p 小 (np ≈ λ)。
For both distributions, you need to calculate probabilities using formulas or tables, and understand when each model is appropriate.
对于这两种分布，你需要使用公式或表格计算概率，并理解何时适用每种模型。
3. Continuous Random Variables & Probability Density Functions | 连续随机变量与概率密度函数
ity density function (PDF), f(x). Probabilities are found by integration: P(a ≤ X ≤ b) = ∫ab f(x) dx. The total area under the curve is 1.
连续随机变量可以在一个区间内取任何值。其概率分布由概率密度函数 (PDF) f(x) 描述。概率通过积分计算：P(a ≤ X ≤ b) = ∫ab f(x) dx。曲线下的总面积为 1。
ies: f(x) ≥ 0 for all x; the cumulative distribution function (CDF) is F(x) = P(X ≤ x) = ∫−∞x dt. The median m satisfies F(m) = 0.5, and the mean is E(X) = ∫ x f(x) dx over the domain.
关键性质：对于所有 x 有 f(x) ≥ 0；累积分布函数 (CDF) 为 F(x) = P(X ≤ x) = ∫−∞x f(t) dt。中位数 m 满足 F(m) = 0.5，均值为 E(X) = ∫ x f(x) dx 在整个定义域上积分。
4. The Normal Distribution & Approximations | 正态分布及其近似
The normal distribution N(μ, σ²) is symmetric and bell-shaped. The standard normal Z ~ μ)/σ. Tables give Φ(z) = P(Z ≤ z). You can handle any normal probability by standardising.
正态分布 N(μ, σ²) 是对称的钟形曲线。标准正态 Z ~ N(0,1) 通过 z = (x − μ)/σ 得到。表格给出 Φ(z) = P(Z ≤ z)。通过标准化可处理任何正态概率。
, using continuity correction. Also, Normal approximates Poisson when λ > 15. Remember to adjust boundaries (0.5 added/subtracted) for discrete approximations.
近似：当 np ≥ 5 且 n(1−p) ≥ 5 时，正态可以近似二项，需使用连续性校正。当 λ > 15 时，正态可近似泊松。记住对于离散近似要调整边界（加/减 0.5）。
5. Hypothesis Testing: Means, Proportions & Correlation | 假设检验：均值、比例与相关
hypothesis H₀. For a population mean with known variance, use the z-test statistic Z = (x̄ − μ) / (σ/√n). For proportions, use Z = (p̂ − p) / √[p(1−p)/n]. Compare with critical values or find p-values.
假设检验评估反对零假设 H₀ 的证据。对于方差已知的总体均值，使用 z 检验统计量 Z = (x̄ − μ) / (σ/√n)。对于比例，使用 Z = (p̂ − p) / √[p(1−p)/n]。与临界值比较或求 p 值
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