Year 13 Cambridge Statistics: Quick Reference Formula & Theorem Handbook | A Level 统计公式定理速查手册

📚 Year 13 Cambridge Statistics: Quick Reference Formula & Theorem Handbook | A Level 统计公式定理速查手册

This handbook provides a concise summary of all the essential formulas, theorems, and critical concepts for the Cambridge Year 13 Statistics syllabus. It is designed as a quick revision tool to reinforce your understanding ahead of examinations. Use it to check definitions, recall distribution properties, and review hypothesis testing procedures.

本手册简明扼要地总结了剑桥 Year 13 统计课程中所有基本公式、定理和重要概念，旨在作为考前快速复习的工具，帮助巩固理解。使用本手册可以快速查阅定义、回顾分布性质并复习假设检验流程。

1. Probability Fundamentals | 概率基础

The probability of an event A is denoted P(A), with 0 ≤ P(A) ≤ 1. For any two events A and B, the addition rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, then P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).

事件 A 的概率记为 P(A)，满足 0 ≤ P(A) ≤ 1。对任意两事件 A 和 B，加法公式为 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 is defined as P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0. Events A and B are independent if and only if P(A ∩ B) = P(A) P(B), which is equivalent to P(A | B) = P(A).

条件概率定义为 P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。事件 A 与 B 相互独立当且仅当 P(A ∩ B) = P(A) P(B)，这也等价于 P(A | B) = P(A)。

Bayes’ theorem allows you to reverse conditional probabilities: P(A | B) = P(B | A) P(A) / P(B). This is especially useful when the probability of the condition is easier to assess from the reverse direction.

贝叶斯定理可用于反转条件概率：P(A | B) = P(B | A) P(A) / P(B)。当反向评估条件概率更容易时，该定理尤为有用。

2. Discrete Random Variables | 离散随机变量

For a discrete random variable X with possible values x_i and probability mass function P(X = x_i), the expectation (mean) is E(X) = μ = ∑ x_i P(X = x_i). The sum runs over all possible values of X.

对于可能取值为 x_i、概率质量函数为 P(X = x_i) 的离散随机变量 X，其期望（均值）为 E(X) = μ = ∑ x_i P(X = x_i)，求和遍及 X 的所有可能值。

The variance of X is Var(X) = E[(X − μ)²] = E(X²) − [E(X)]². The standard deviation is σ = √Var(X). Linear transformations obey E(aX + b) = a E(X) + b and Var(aX + b) = a² Var(X) for constants a and b.

X 的方差为 Var