📚 A-Level CAIE Statistics: Formula and Theorem Quick Reference Handbook | A-Level CAIE 统计：公式定理速查手册
This handbook summarises the key formulae and theorems tested in CAIE A-Level Mathematics Paper 5 (Probability & Statistics 1) and Paper 6 (Probability & Statistics 2). Use it as a quick revision checklist and exam reference.
本手册汇总 CAIE A-Level 数学 Paper 5（概率与统计 1）和 Paper 6（概率与统计 2）中的核心公式与定理，可作为复习清单和考前速查使用。
1. Data Representation and Summary Statistics | 数据表示与汇总统计
For raw data, the mean and variance are calculated from the sum of values and the sum of squared values.
x̄ = Σx / n, s² = Σ(x − x̄)² / n = Σx² / n − x̄², s = √s²
对于原始数据，均值和方差分别通过数值总和与平方值总和计算，标准差为方差的平方根。
For grouped data, use class midpoints m and frequencies f.
x̄ = Σfm / Σf, s² = Σfm² / Σf − x̄²
对于分组数据，使用组中值 m 和频数 f 进行计算。
ded as y = (x − a) / b, the original mean and standard deviation are recovered as follows.
y = (x − a) / b, x̄ = a + bȳ, sₓ = |b| sy
若数据以 y = (x − a) / b 进行编码，原始均值与标准差按上述公式还原。
2. Probability Rules | 概率法则
The general addition rule handles overlapping events.
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
一般加法公式用于处理有重叠事件的情况。
If A and B are mutually exclusive, their intersection has probability zero.
P(A ∩ B) = 0
若事件 A 与 B 互斥，则它们的交事件概率为零。
Conditional probability and independence are defined by the following relationships.
P(A | B) = P(A ∩ B) / P(B), P(A ∩ B) = P(A)P(B) if independent
条件概率和独立性的定义如上；若 A 与 B 独立，则交事件概率等于各自概率的乘积。
3. Permutations and Combinations | 排列与组合
The number of ordered arrangements of r objects chosen from n distinct objects is given by nPr.
n! = n(n−1)(n−2)⋯1, nPr = n! / (n − r)!
从 n 个不同对象中有序选取 r 个的排列数由 nPr 给出。
The number of unordered selections is given by nCr.
nCr = n! / [r!(n − r)!]
从 n 个不同对象中无序选取 r 个的组合数由 nCr 给出。
For arrangements with repeated items, divide by the factorial of each repeated group size.
Number of arrangements = n! / (p! q! ⋯)
当排列中存在重复对象时，需除以各重复组大小的阶乘。
4. Discrete Random Variables | 离散随机变量
The expectation of a discrete random variable is the probability-weighted average of its values.
E(X) = Σ x P(X = x) = μ
离散随机变量的期望是其取值按照概率加权的平均值。
The variance can be computed using the second moment about the origin.
Var(X) = E(X²) − [E(X)]² = Σ x² P(X = x) − μ²
方差可使用二阶原点矩减去期望的平方来计算。
Linear transformations affect expectation and variance as follows.
E(aX + b) = aE(X) + b, Var(aX + b) = a² Var(X)
线性变换对期望和方差的影响如上所示；方差中常数项 b 不影响波动。
5. Binomial Distribution | 二项分布
If X ~ B(n, p), the probability of exactly r successes in n independent trials is given by the binomial formula.
P(X = r) = nCr pʳ (1 − p)ⁿ⁻ʳ
若 X ~ B(n, p)，在 n 次独立试验中恰好出现 r 次成功的概率由二项式公式给出。
mean and variance of a binomial distribution are simple multiples of n, p, and q = 1 − p.
E(X) = np, Var(X) = npq, q = 1 − p
二项分布的期望为 np，方差为 npq，其中 q = 1 − p。
The binomial model requires a fixed number of trials, constant success probability, and independent trials.
二项模型要求试验次数固定、每次成功概率不变，并且各次试验相互独立。
6. Geometric Distribution | 几何分布
If X ~ Geo(p), X counts the number of trials up to and including the first success.
P(X = r) = (1 − p)ʳ⁻¹ p, r = 1, 2, 3, ⋯
若 X ~ Geo(p)，则 X 表示首次成功出现时已经进行的试验次数，其概率质量函数如上。
The mean and variance of a geometric distribution are based on the success probability p.
E(X) = 1 / p, Var(X) = q / p², q = 1 − p
几何分布的期望为 1/p，方差为 q/p²，其中 q = 1 − p。
The geometric distribution is memoryless: past failures do not change the probability of future success.
几何分布具有无记忆性：过去的失败不会改变未来成功的概率。
7. Normal Distribution | 正态分布
If X ~ N(μ, σ²), the standardised score converts X to the standard normal variable Z.
Z = (X − μ) / σ, X ~ N(μ, σ²), Z ~ N(0, 1)
若 X ~ N(μ, σ²)，则将 X 标准化得到标准正态变量 Z，便于查表计算概率。
When a discrete distribution is approximated by a normal distribution, apply continuity correction.
P(X ≤ r) ≈ P(Y < r + 0.5), P(X ≥ r) ≈ P(Y > r − 0.5)
当用正态分布近似离散分布时，需要使用连续性修正以提高精度。
To find an unknown mean or standard deviation, use the inverse normal function on the standardised equation.
如需反求未知均值或标准差，可对标准化方程使用逆正态函数求解。
8. Poisson Distribution | 泊松分布
If X ~ Po(λ), the probability of exactly r occurrences in a fixed interval is given by the Poisson formula.
P(X = r) = e−λ × λ^r / r!, r = 0, 1, 2, ⋯
若 X ~ Po(λ)，在固定区间内恰好发生 r 次事件的概率由泊松公式给出。
The Poisson distribution has equal mean and variance.
E(X) = λ, Var(X) = λ
泊松分布的期望和方差相等，均为 λ。
If X and Y are independent Poisson variables, their sum is also Poisson with mean equal to the sum of the means.
X ~ Po(λ), Y ~ Po(μ), X + Y ~ Po(λ + μ)
若 X 与 Y 相互独立且均服从泊松分布，则 X + Y 仍服从泊松分布，参数为 λ + μ。
9. Continuous Random Variables | 连续随机变量
For a continuous random variable with probability density function f(x), probabilities are found by integration.
P(a < X < b) = ∫ₐᵇ f(x) dx, ∫ f(x) dx = 1 over the domainPublished by TutorHao | A-Level 统计 Revision Series | aleveler.com
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