A-Level Mathematics: Fundamental Concepts and Representations of Probability Distributions | A-Level 数学：概率分布的基本概念与表示

📚 A-Level Mathematics: Fundamental Concepts and Representations of Probability Distributions | A-Level 数学：概率分布的基本概念与表示

Probability distributions lie at the heart of statistics and probability theory. In A-Level Mathematics, you are expected not only to calculate probabilities but also to understand how a random variable is described by its distribution, whether through a table, a graph, or a formula. This article presents the essential definitions, representations, and properties you need to master.

概率分布是统计学与概率论的核心。在A-Level数学中，你不仅需要计算出概率，更要理解随机变量如何通过表格、图形或公式来被其分布所描述。本文系统梳理你必需掌握的基本定义、表示方式与核心性质。

1. Random Variables and Their Role | 随机变量及其作用

A random variable is a variable whose possible values are the numerical outcomes of a random experiment. We usually denote it by a capital letter such as X, Y or Z. For example, if we count the number of heads in three tosses of a coin, the random variable X can take the values 0, 1, 2, 3.

随机变量是取值由随机试验的数值结果所决定的变量。我们通常用大写字母如 X、Y 或 Z 来表示。例如，投掷一枚硬币三次，记录出现正面的次数，随机变量 X 可取值 0、1、2、3。

A random variable maps outcomes from the sample space to real numbers.

随机变量将样本空间中的结果映射到实数。
Once we assign probabilities to each value, we obtain a probability distribution.

当我们为每一个取值赋予概率后，便得到概率分布。

2. Discrete and Continuous Distributions | 离散分布与连续分布

Random variables are classified into two broad types: discrete and continuous. A discrete random variable takes a countable set of values, often whole numbers. A continuous random variable takes any value within an interval, such as height or time. This distinction determines how probabilities are represented.

随机变量分为两大类：离散型与连续型。离散随机变量的取值是可数的，通常是整数；连续随机变量则可以取某个区间内的任意值，例如身高或时间。这一区分决定了概率的表示方式。

Discrete distributions: probabilities are concentrated at individual points.

离散分布：概率集中在各个单独的点上。
Continuous distributions: probabilities are represented by areas under a curve.

连续分布：概率由曲线下的面积来表示。
For a continuous variable, P(X = a) = 0 for any exact value a.

对于连续变量，任何一个精确取值 a 的 P(X = a) = 0。

3. Probability Mass Function (PMF) | 概率质量函数

For a discrete random variable X, the probability mass function (PMF) gives the probability that X takes each particular value. It is written as p(x) = P(X = x). The PMF satisfies two key conditions:

对于离散随机变量 X，概率质量函数（PMF）给出 X 取各个特定值的概率，记作 p(x) = P(X = x)。PMF 满足两个关键条件：

p(x) ≥ 0 for all x, and Σ p(x) = 1

Consider tossing two fair coins and letting X be the number of heads. The distribution is shown below.

考虑抛掷两枚均匀硬币，令 X 为出现正面的次数，其分布如下表所示。

x	0	1	2
P(X = x)	¼	½	¼

Check that ¼ + ½ + ¼ = 1. The PMF can be displayed in a vertical line chart, with one line for each value.

验证 ¼ + ½ + ¼ = 1。PMF 可用垂直线状图展示，每个取值画一条线段。

4. Probability Density Function (PDF) | 概率密度函数

For a continuous random variable X, probabilities are described by a probability density function (PDF), denoted f(x). The PDF must satisfy f(x) ≥ 0 for all x, and the total area under the curve over the whole range equals 1.

对于连续随机变量 X，概率由概率密度函数（PDF）描述，记作 f(x)。PDF 必须满足对所有 x 有 f(x) ≥ 0，且曲线下方在整个取值范围内的总面积为 1。

∫ f(x) dx = 1 (over the entire range of x)

To find P(a < X < b), we calculate the area under the curve between x = a and x = b:

求 P(a < X < b) 时，我们计算曲线在 x = a 与 x = b 之间的面积：

P(a < X < b) = ∫ₐᵇ f(x) dx

For example, the uniform distribution on [0, 1] has f(x) = 1 for 0 ≤ x ≤ 1 and f(x) = 0 elsewhere. Then P(0.25 < X < 0.75) = 0.5. Because exact values have zero probability, we never write P(X = k) for a continuous variable.

例如，[0, 1] 上的均匀分布有 f(x) = 1（当 0 ≤ x ≤ 1），其余为 0。于是 P(0.25 < X < 0.75) = 0.5。由于精确取值的概率为零，我们从不写连续变量的 P(X = k)。

5. Cumulative Distribution Function (CDF) | 累积分布函数

The cumulative distribution function F(x) is defined as F(x) = P(X ≤ x). It accumulates probability up to the value x and is a non-decreasing function. For a discrete variable, F(x) is a step function; for a continuous variable, F(x) is continuous and increases smoothly from 0 to 1.

累积分布函数 F(x) 定义为 F(x) = P(X ≤ x)。它累积到取值 x 为止的概率，且是不减函数。对于离散变量，F(x) 是阶梯函数；对于连续变量，F(x) 连续且从 0 平滑增加到 1。

Key relationships for the CDF are:

CDF 的关键关系如下：

For continuous X: F′(x) = f(x).

对于连续变量 X：F′(x) = f(x)。
P(a < X ≤ b) = F(b) − F(a).

P(a < X ≤ b) = F(b) − F(a)。
F(x) → 0 as x → −∞, and F(x) → 1 as x → +∞.

当 x → −∞ 时 F(x) → 0，当 x → +∞ 时 F(x) → 1。

6. Mean and Variance of a Distribution | 分布的均值与方差

The mean (expectation) E(X) measures the centre of a distribution, while the variance Var(X) measures its spread. Both can be computed from the PMF or PDF.

均值（期望）E(X) 衡量分布的中心位置，方差 Var(X) 衡量分布的离散程度。两者都可以由 PMF 或 PDF 计算得到。

For a discrete random variable:

对于离散随机变量：

E(X) = Σ x p(x), Var(X) = Σ (x − μ)² p(x) = E(X²) − μ²

For a continuous random variable:

对于连续随机变量：

E(X) = ∫ x f(x) dx, Var(X) = ∫ (x − μ)² f(x) dx

The standard deviation is σ = √Var(X). Two useful rules for linear transformations are:

标准差为 σ = √Var(X)。两个常用的线性变换规则是：

E(aX + b) = aE(X) + b, Var(aX + b) = a²Var(X)

Notice that adding a constant does not change the variance, but multiplying by a constant scales the variance by a².

注意：加上常数不改变方差，而乘以常数 a 会使方差变成原来的 a² 倍。

7. Standard Discrete Distributions: Binomial and Poisson | 标准离散分布：二项分布与泊松分布

Two discrete distributions appear constantly in A-Level work: the binomial distribution and the Poisson distribution.

两个在A-Level中频繁出现的离散分布是：二项分布与泊松分布。

If X ~ B(n, p) counts the number of successes in n independent trials, each with success probability p, then

若 X ~ B(n, p) 表示 n 次独立试验中成功的次数，每次成功概率为 p，则

P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ, E(X) = np, Var(X) = np(1 − p)

If X ~ Po(λ) models the number of rare events occurring in a fixed interval, with mean rate λ, then

若 X ~ Po(λ) 建模固定区间内稀有事件发生的次数，平均速率为 λ，则

P(X = r) = e⁻λ λʳ / r!, E(X) = λ, Var(X) = λ

The Poisson distribution is often used when events occur independently and at a constant average rate. When n is large and p is small, the binomial distribution may be approximated by Po(np).

泊松分布常用于事件独立发生且平均速率恒定的情形。当 n 很大、p 很小时，二项分布可以用 Po(np) 近似。

8. The Normal Distribution | 正态分布

The normal distribution is the most important continuous distribution. If X ~ N(μ, σ²), its PDF is bell-shaped, symmetric about μ, and its mean, median and mode are all equal. The total area under the curve is 1.

正态分布是最重要的连续型分布。若 X ~ N(μ, σ²)，其 PDF 呈钟形，关于 μ 对称，且均值、中位数、众数均相等。曲线下总面积为 1。

To find probabilities, we standardise X to the standard normal variable Z:

为求概率，我们将 X 标准化为标准正态变量 Z：

Z = (X − μ) / σ ~ N(0, 1)

Then P(X < a) = P(Z < (a − μ) / σ). The standard normal table gives Φ(z) = P(Z ≤ z). Because the distribution is symmetric, Φ(−z) = 1 − Φ(z).

于是 P(X < a) = P(Z < (a − μ) / σ)。标准正态表给出 Φ(z) = P(Z ≤ z)。由于分布对称，Φ(−z) = 1 − Φ(z)。

Approximately 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three. This rule is useful for quick estimates.

约 68% 的数据位于均值的一个标准差内，95% 位于两个标准差内，99.7% 位于三个标准差内。该经验法则可用于快速估计。

9. Using Distribution Tables and Calculators | 分布表与计算器的使用

In examinations, you are expected to use statistical tables or a calculator to evaluate probabilities for standard distributions. Always check whether the table gives cumulative probabilities P(X ≤ r) or tail probabilities P(X > r).

考试中，你需要使用统计表或计算器来计算标准分布的概率。始终检查所给表格是累积概率 P(X ≤ r) 还是尾部概率 P(X > r)。

Binomial: use tables for selected n and p, or calculate directly using the PMF formula.

二项分布：对给定的 n 和 p 查表，或直接用 PMF 公式计算。
Poisson: tabulated values usually give cumulative probabilities; for P(X = r), subtract two consecutive cumulative values.

泊松分布：表中通常是累积概率；求 P(X = r) 时用两个相邻累积值相减。
Normal: always standardise first; be careful with inequalities such as P(X ≥ a) = 1 − P(X < a).

正态分布：先标准化，注意不等号方向，如 P(X ≥ a) = 1 − P(X < a)。

When using the normal distribution to approximate a binomial distribution, apply a continuity correction. For example, P(X ≤ k) becomes P(Y ≤ k + 0.5) where Y is the approximating normal variable.

使用正态分布近似二项分布时，应作连续性修正。例如，P(X ≤ k) 变为 P(Y ≤ k + 0.5)，其中 Y 是近似正态变量。

10. Common Exam Pitfalls | 常见考试陷阱

Many marks are lost in probability distribution questions because of small misunderstandings. The following pitfalls are especially common:

在概率分布题目中，许多分数因为小误解而失掉。以下几个陷阱尤其常见：

Using a PDF formula such as ∫ f(x) dx for a discrete variable; always match the representation to the type of variable.

对离散变量误用 PDF 的积分公式；务必使表示方式与变量类型匹配。
Forgetting that the total probability is 1, and omitting to check whether a given PMF or PDF is valid.

忘记总概率为 1，未检验给定的 PMF 或 PDF 是否有效。
Mixing up Var(X) = E(X²) − (E(X))² and writing E(X²) − E(X); the square is essential.

混用 Var(X) = E(X²) − (E(X))²，写成 E(X²) − E(X)；平方必不可少。
Using the wrong tail of the normal distribution; draw a sketch to confirm the region you need.

使用正态分布的错误尾端；画草图来确定所需区域。
Omitting the continuity correction when approximating a discrete distribution by a continuous one.

用连续分布近似离散分布时省略连续性修正。

Mastering these concepts will allow you to handle probability distribution questions with confidence. Always begin by identifying the type of random variable, then choose the correct representation and apply the relevant formulas carefully.

掌握这些概念将使你能够自信地处理概率分布题目。做题时先判断随机变量的类型，再选择正确的表示方式，并仔细应用相关公式。

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A-Level Mathematics: Fundamental Concepts and Representations of Probability Distributions | A-Level 数学：概率分布的基本概念与表示