Formula & Theorem Quick Reference for AQA Year 13 Statistics | Year 13 AQA 统计：公式定理速查手册

📚 Formula & Theorem Quick Reference for AQA Year 13 Statistics | Year 13 AQA 统计：公式定理速查手册

This quick reference handbook compiles the most important formulas and theorems for the AQA Year 13 Statistics syllabus, serving as a concise revision tool that covers probability, distributions, inference, and regression. Each entry is presented as a centre-stage formula followed by a bilingual explanation to reinforce understanding.

本速查手册汇集了AQA Year 13 统计课程中的核心公式与定理，是一份覆盖概率、分布、推断和回归的简洁复习资料。每个条目均以居中公式展示，并配以中英文双语解释，帮助巩固理解。

1. Probability Basics | 概率基础

Probability rules form the bedrock of statistical inference. The key identities govern how we combine events, condition on information, and check for independence.

概率法则是统计推断的基石。以下核心恒等式描述了事件的组合方式、条件信息的处理以及独立性的检验。

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Addition rule: the probability of the union of two events is the sum of their individual probabilities minus the probability of their intersection.

加法公式：两个事件并集的概率等于各自概率之和减去交集概率。

P(A | B) = P(A ∩ B) / P(B), P(B) > 0

Conditional probability: the likelihood of A given that B has occurred. It refines the sample space to B.

条件概率：在事件B已发生的条件下事件A的概率，将样本空间缩减为B。

P(A ∩ B) = P(A | B) P(B) = P(B | A) P(A)

Multiplication rule: joint probability can be expressed using conditional probabilities in either order.

乘法法则：联合概率可用任意顺序的条件概率乘积表示。

Independence: P(A ∩ B) = P(A) P(B) ⇔ P(A | B) = P(A)

Two events are independent if the occurrence of one does not affect the probability of the other.

若事件A与B相互独立，则一个事件的发生不影响另一个事件的发生概率。

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2. Bayes' Theorem | 贝叶斯定理

Bayes' theorem reverses conditional probabilities, updating prior beliefs with new evidence. It is particularly useful when the sample space is partitioned.

贝叶斯定理用于逆转条件概率，依据新证据更新先验信念，在样本空间可划分时尤为有用。

P(A | B) = [P(B | A) P(A)] / P(B)

The basic form of Bayes' theorem, expressing the posterior probability of A given B.

贝叶斯定理的基本形式，给出了在B发生的条件下A的后验概率。

P(A | B) = [P(B | A) P(A)] / [P(B | A) P(A) + P(B | A') P(A')]

Full version using the law of total probability for a partition {A, A'}.

使用全概率公式的完整版本，基于划分 {A, A'} 计算。

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3. Discrete Random Variables | 离散随机变量

Discrete random variables take countable values. Their behaviour is summarised by the probability mass function (PMF), expectation, and variance.

离散随机变量取可数值，其行为由概率质量函数、期望和方差概括。

E(X) = Σ x P(X = x)

Expected value (mean) of a discrete random variable: the probability-weighted average of all possible values.

离散随机变量的期望（均值）：所有可能取值与其概率乘积之和。

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

Variance measures spread. The computational formula E(X²) – μ² is often easier to evaluate.

方差衡量离散程度，常用计算式 E(X²) – μ² 更易于求值。

E(aX + b) = a E(X) + b

Linearity of expectation: scaling and shifting transform the mean accordingly.

期望的线性性质：缩放和平移按同样规则变换均值。

Var(aX + b) = a² Var(X)

Variance is affected by scaling (squared) but not by a shift, because adding a constant does not change spread.

方差受缩放影响（带平方），但平移不改变方差，因为加常数不改变离散程度。

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4. Binomial Distribution | 二项分布

The binomial model counts successes in a fixed number of independent Bernoulli trials. It is determined by the number of trials n and success probability p.

二项分布用于统计固定次数的独立伯努利试验中的成功次数，由试验次数 n 和成功概率 p 决定。

X ~ B(n, p): P(X = k) = ⁿCₖ pᵏ (1 – p)ⁿ⁻ᵏ

Probability of exactly k successes in n trials. ⁿCₖ is the binomial coefficient.

n 次试验中恰好成功 k 次的概率，ⁿCₖ 为二项式系数。

E(X) = np

Mean of the binomial distribution.

二项分布的均值。

Var(X) = np(1 – p)

Variance of the binomial distribution.

二项分布的方差。

Conditions: trials are independent; fixed n; constant p; two outcomes.

Binomial conditions must be satisfied for the model to be valid.

使用二项分布需满足试验独立、次数固定、概率恒定、仅两种结果的条件。

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5. Poisson Distribution | 泊松分布

The Poisson distribution models the number of events occurring in a fixed interval of time or space when events happen independently at a constant average rate λ.

泊松分布用于建模在固定时间或空间间隔内独立发生的事件数，事件以恒定平均速率 λ 发生。

X ~ Po(λ): P(X = k) = (λᵏ e⁻λ) / k!

Probability mass function of a Poisson random variable.

泊松随机变量的概率质量函数。

E(X) = λ, Var(X) = λ

Mean and variance are both equal to the rate parameter λ.

泊松分布的均值和方差都等于参数 λ。

Poisson approximation to Binomial: X ~ B(n, p) with n large, p small, np = λ → X approx Po(λ)

arge and p is small, the binomial distribution can be approximated by Poisson with λ = np.

当 n 很大而 p 很小时，可用参数 λ = np 的泊松分布近似二项分布。

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6. Continuous Random Variables & PDF | 连续随机变量与概率密度函数

Continuous random variables are described by probability density functions (PDFs) and cumulative distribution functions (CDFs). Probabilities correspond to areas under the PDF.

连续随机变量用概率密度函数和累积分布函数描述，概率对应密度曲线下的面积。

f(x) ≥ 0, ∫₋∞⁺∞ f(x) dx = 1

Conditions for a valid PDF: non-negative and integrates to 1 over the whole range.

有效概率密度函数的条件：非负且在整个定义域上积分为 1。

F(x) = P(X ≤ x) = ∫₋∞ˣ f(t) dt

Cumulative distribution function gives the probability up to x.

累积分布函数给出随机变量不超过 x 的概率。

E(X) = ∫ x f(x) dx, Var(X) = ∫ x² f(x) dx – μ²

Mean and variance for continuous distributions use integration instead of summation.

连续分布的均值与方差通过积分而非求和计算。

Rectangular (Uniform) on [a, b]: f(x) = 1/(b – a), E(X) = (a + b)/2, Var(X) = (b – a)² / 12

Special case: the continuous uniform distribution. Every subinterval of equal length has the same probability.

特例：连续均匀分布，等长子区间概率相同。

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7. Normal Distribution | 正态分布

The normal distribution is the most important continuous model in statistics. Its bell-shaped curve is defined by the mean μ and variance σ².

正态分布是统计学中最重要的连续模型，其钟形曲线由均值 μ 和方差 σ² 决定。 Published by TutorHao | Year 13 统计 Revision Series | aleveler.com

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