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 gathers every essential formula, theorem and definition you need for the Cambridge Year 13 Statistics modules into one compact revision resource. Each section pairs the statement of the result with the conditions under which it may be applied, because examiners award marks for correct justification as well as correct arithmetic.

本手册把剑桥 Year 13 统计模块所需的全部核心公式、定理与定义汇总为一份紧凑的复习资料。每一节不仅给出结论本身，还说明其适用条件，因为考官既看计算是否正确，也看理由是否充分。

---

1. Probability Fundamentals and Set Notation | 概率基础与集合记号

P(A) ≤ 1. The whole sample space S satisfies P(S) = 1, and the complement of A, written A', has probability P(A') = 1 − P(A).

事件 A 的概率取值在 0 与 1 之间（含端点），即 0 ≤ P(A) ≤ 1。整个样本空间 S 满足 P(S) = 1，A 的补事件记作 A'，其概率为 P(A') = 1 − P(A)。

n. This counting definition underlies every combinatorial probability question in Paper 5.

若样本空间由 n 个等可能结果组成，事件 A 包含其中 k 个，则 P(A) = k / n。这一计数定义是 Paper 5 中所有组合概率题的根基。

The addition rule for two events is the formula you will use most often:

两事件的加法公式是使用频率最高的公式：

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

If A and B are mutually exclusive they cannot occur together, so P(A ∩ B) = 0 and the rule reduces to P(A ∪ clusive is not the same as independent, and confusing the two is a very common exam error.

若 A 与 B 互斥，则二者不可能同时发生，故 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。注意互斥与独立是两个不同的概念，混淆二者是极常见的考试错误。

Set notation appears throughout the paper, so learn to translate it quickly: A ∪ B means 'A or B or both'; A ∩ B means 'A and B'; A' means 'not A'; and A ∩ B' means 'A but not B'.

集合记号贯穿整张试卷，需要熟练互译：A ∪ B 表示 "A 或 B 或两者"；A ∩ B 表示 "A 且 B"；A' 表示 "非 A"；A ∩ B' 表示 "A 但非 B"。

---

2. Conditional Probability, Independence and Bayes' Theorem | 条件概率、独立性与贝叶斯定理

The conditional probability of A given that B has occurred is defined by the quotient below, and the definition requires P(B) > 0.

在 B 已发生的条件下，A 的条件概率由下面的商式定义，且要求 P(B) > 0。

P(A | B) = P(A ∩ B) / P(B)

Rearranging gives the multiplication rule P(A ∩ B) = P(B) × P(A | B) = P(A) × P(B | A). This symmetric form is exactly what makes Bayes' theorem possible, and it is the fastest route into most tree-diagram problems.

移项即得乘法公式 P(A ∩ B) = P(B) × P(A | B) = P(A) × P(B | A)。这种对称形式正是贝叶斯定理成立的原因，也是解决大多数树状图问题最快的路径。

Two 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) and also to P(B | A) = P(B). All three statements are interchangeable, so use whichever is easiest to test.

两事件 A 与 B 相互独立当且仅当 P(A ∩ B) = P(A) × P(B)，这等价于 P(A | B) = P(A)，也等价于 P(B | A) = P(B)。三种表述可以互换，检验时选择最方便的一种即可。

Independence does not carry over to complements in the obvious direction: if A and B are independent, then A and B' are also independent, and so are A' and B'. Mutually exclusive events, by contrast, are dependent unless one of them has probability zero.

独立性对补事件同样成立：若 A 与 B 独立，则 A 与 B' 也独立，A' 与 B' 亦然。相反，互斥事件通常是相关的，除非其中一个事件概率为零。

Bayes' theorem reverses the conditioning, allowing you to update a prior probability in the light of new evidence.

贝叶斯定理用于反转条件方向，使你能根据新证据更新先验概率。

P(A | B) = P(B | A) × P(A) / P(B)

ften unknown and must itself be decomposed, using the law of total probability: P(B) = P(B | A) P(A) + P(B | A') P(A' e diagram first, label every branch, then multiply along the branches and add across them.

实际做题时 P(B) 常常未知，需要用全概率公式展开：P(B) = P(B | A) P(A) + P(B | A') P(A')。先画树状图，标注每条分支，然后沿分支相乘、横向相加。

---

3. Permutations, Combinations and Counting | 排列、组合与计数

The factorial n! counts the arrangements of n distinct objects in a row. The number of ordered selections of r objects from n is the permutation

阶乘 n! 表示 n 个不同对象排成一列的排列数。从 n 个对象中有序选取 r 个的数目称为排列数

ⁿPᵣ = n! / (n − r)!

and the number of unordered selections, where order does not matter, is the binomial coefficient

而无序选取的数目（顺序无关）称为二项系数

ⁿCᵣ = n! / (r! (n − r)!)

The phrase 'at least' . Calculating the complement is usually far quicker than summing many terms, and it is one of the highest-value shortcuts in the paper.

广告 / Sponsored

"至少" 一词几乎总意味着用补事件：P(至少一个) = 1 − P(一个都没有)。计算补事件通常比逐项相加快得多，是整张试卷性价比最高的技巧之一。

When some objects are identical, divide by the factorial of each repeated group. For example, the arrangements of the letters of a word with repeated letters are counted by n! / (a! b! ...), where a, b, ... are the frequencies of the repeated letters.

当对象中有重复时，需除以每组重复对象的阶乘。例如含有重复字母的单词排列数为 n! / (a! b! …)，其中 a、b、… 为各重复字母的出现次数。

The binomial coefficients ⁿCᵣ are the same numbers that appear in the binomial expansion and in the binomial probability formula, so fluency with combinations pays off twice: in counting questions and in distribution questions.

二项系数 ⁿCᵣ 同时出现在二项展开式与二项概率公式中，因此熟练组合数会带来双重回报：既能应付计数题，也能应付分布题。

---

4. Discrete Random Variables: Expectation and Variance | 离散随机变量：期望与方差

1 over all possible values. The expectation, or mean, is the probability-weighted average

离散随机变量 X 取值为 xᵢ，对应概率为 P(X = xᵢ)，这些概率必须满足 Σ P(X = xᵢ) = 1（求和遍及所有可能取值）。期望（均值）即概率加权平均

E(X) = μ = Σ xᵢ P(X = xᵢ)

(g(X)) = Σ g(xᵢ) P(X = xᵢ). This is the expected-value version of the transformation rule.

对于随机变量的函数 g，无需先求出 g(X) 的分布即可套用同一原理：E(g(X)) = Σ g(xᵢ) P(X = xᵢ)。这是期望的变换法则。

The variance measures spread about the mean and has two equivalent computational forms:

方差衡量围绕均值的离散程度，有两种等价的计算形式：

Var(X) = E[(X − μ)²] = E(X²) − [E(X)]²

tabulated. The standard deviation is σ = √Var(X) and is expressed in the same units as X.

第二种形式在考试中通常更快，因为它只需用到你已经列出的原点矩 E(X) 与 E(X²)。标准差为 σ = √Var(X)，单位与 X 相同。

Linear transformations obey two rules that must be memorised exactly, for constants a and b:

对常数 a 与 b，线性变换遵循两条必须牢记的法则：

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

Notice that adding a constant shifts the mean but leaves the variance unchanged, while multiplying by a scales the standard deviation by |a| rather than by a. A negative multiplier flips the distribution but cannot make the variance negative.

注意：加常数只平移均值而不改变方差；乘以系数会使标准差按 |a| 缩放，而不是按 a 缩放。负数乘子会翻转分布，但方差永远非负。

tained by adding the probabilities of all values up to and including x₀. Most questions on 'at most', 'fewer than' and 'no more than' are answered directly from F.

累积分布函数为 F(x₀) = P(X ≤ x₀)，即把不超过 x₀ 的所有取值的概率相加。凡出现 "至多"、"少于"、"不超过" 的题目，基本都可直接由 F 作答。

---

5. The Binomial Distribution | 二项分布

If X counts the number of successes in n independent trials, each with the same success probability p, then X follows a binomial distribution, written X ~ B(n, p), with probability mass function

若 X 表示 n 次独立试验中成功的次数，每次成功的概率同为 p，则 X 服从二项分布，记作 X ~ B(n, p)，其概率质量函数为

P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ    for r = 0, 1, 2, …, n

The four conditions to state are: a fixed number of trials n; only two possible outcomes per trial; a constant probability p; and independence between trials. If any condition fails, the binomial model is not valid and a mark will be lost for failing to notice.

使用前需说明四个条件：试验次数 n 固定；每次试验只有两种结果；成功概率 p 恒定；各次试验相互独立。任一条件不满足，二项模型即不适用，若未察觉就会失分。

The mean and variance follow directly from the parameters and need no summation:

均值与方差可由参数直接得到，无需求和：

E(X) = np    and    Var(X) = np(1 − p)

ar P(X ≥ r) = 1 − P(X ≤ r − 1) and P(X < r) = P(X ≤ r − 1), while P(X > r) = 1 − P(X ≤ r).

累积概率可查表，但表格给出的是 P(X ≤ r)，其他形式都要转换。特别地，P(X ≥ r) = 1 − P(X ≤ r − 1)，P(X < r) = P(X ≤ r − 1)，而 P(X > r) = 1 − P(X ≤ r)。

The binomial distribution is also the model behind hypothesis tests on a proportion, where the null hypothesis fixes p₀ and the observed count r is compared with the critical region from the tables.

二项分布也是比例假设检验的模型：原假设给定 p₀，再将观测次数 r 与查表得到的临界域比较。

---

6. The Poisson Distribution | 泊松分布

If X

更多咨询请联系16621398022（同微信）