Continuous Distributions | 连续分布

📚 Continuous Distributions | 连续分布

Continuous distributions form a fundamental part of the Edexcel A-Level Statistics syllabus. Unlike discrete random variables that take countable values, continuous random variables can assume any value within an interval, and their probabilities are described by areas under smooth curves. In this article, we will explore probability density functions, cumulative distribution functions, expectation, variance, and the two most important continuous distributions: the uniform and the normal distribution. Mastering these concepts is essential for tackling real-world data modelling and exam questions confidently.

连续分布是爱德思 A-Level 统计课程的核心内容。与取可数值的离散随机变量不同，连续随机变量可以在某个区间内取任意值，概率由光滑曲线下的面积来描述。本文将深入探讨概率密度函数、累积分布函数、期望、方差，以及最重要的两种连续分布：均匀分布和正态分布。掌握这些概念对于自信地应对真实数据建模和考试题目至关重要。

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1. What is a Continuous Random Variable? | 什么是连续随机变量？

A continuous random variable X can take any real value within a given range. For example, the height of a randomly selected student, the time taken to complete a task, or the volume of liquid in a bottle are all continuous quantities. We cannot list all possible outcomes as we do for a discrete variable, so we describe the distribution using a probability density function (PDF) and calculate probabilities as areas under the curve.

连续随机变量 X 可以在给定范围内取任意实数值。例如，随机选择的一名学生身高、完成一项任务所用的时间或瓶中液体的体积都是连续量。我们无法像离散变量那样列出所有可能的结果，因此我们用概率密度函数 (PDF) 描述分布，并将概率计算视为曲线下的面积。

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

For a continuous random variable X, the probability density function f(x) describes the relative likelihood of X taking a value near x. The key property is that the probability of X lying in an interval [a, b] is given by the definite integral:

P(a ≤ X ≤ b) = ∫_a^b f(x) dx

对于连续随机变量 X，概率密度函数 f(x) 描述 X 在 x 附近取值的相对可能性。关键性质是 X 落在区间 [a, b] 的概率由定积分给出：

P(a ≤ X ≤ b) = ∫_a^b f(x) dx

because the area above a point is zero. Probabilities are only meaningful over intervals.

务必记住，对于任意单点 c，P(X = c) = 0，因为单个点上的面积为零。概率仅在区间上才有意义。

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3. Valid PDF Conditions | PDF 的有效条件

domain: (i) f(x) ≥ 0 for all x, and (ii) the total area under the curve equals 1, i.e., ∫_-∞^∞ These ensure that probabilities are non-negative and that the total probability sums to 1.

一个函数 f(x) 能作为有效的 PDF，必须在其定义域上满足两个条件：(i) 对所有 x 有 f(x) ≥ 0；(ii) 曲线下的总面积为 1，即 ∫_-∞^∞ f(x) dx = 1。这确保了概率非负且总概率之和为 1。

In exam questions, you may be asked to find the value of an unknown constant k that makes f(x) a valid PDF by setting the integral equal to 1 and solving for k.

在考试题目中，你可能会被要求求出使 f(x) 成为有效 PDF 的未知常数 k，方法是将积分设为 1 并解出 k。

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

The cumulative distribution function, denoted by F(x), gives the probability that X takes a value less than or equal to x. It is defined as:

F(x) = P(X ≤ x) = ∫_-∞^x f(t) dt

累积分布函数，记作 F(x)，表示 X 取值小于或等于 x 的概率。其定义为：

F(x) = P(X ≤ x) = ∫_-∞^x f(t) dt

nd 0 as x → −∞. Furthermore, the PDF can be recovered by differentiating the CDF: f(x) = F' is extremely useful for moving between probability expressions and distribution functions.

CDF 是一个非递减函数，当 x → ∞ 时趋近于 1，x → −∞ 时趋近于 0。此外，通过对 CDF 求导可以恢复 PDF：f(x) = F'(x)，前提是导数存在。这一关系在概率表达式与分布函数之间切换时极为有用。

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5. Expectation and Variance | 期望与方差

a continuous random variable X with PDF f(x), the expectation (mean) μ = E(X) is given by:

E(X) = ∫ x f(x) dx

对于具有 PDF f(x) 的连续随机变量 X，期望（均值）μ = E(X) 由下式给出：

E(X) = ∫ x f(x) dx

The variance σ² = Var(X) can be calculated using the formula:

Var(X) = E(X²) − [E(X)]² = ∫ x² f(x) dx − μ²

方差 σ² = Var(X) 可用以下公式计算：

Var(X) = E(X²) − [E(X)]² = ∫ x² f(x) dx − μ²

These integrals are evaluated over the entire range of X. Expectation and variance retain their usual interpretations: the mean is the central location, and the variance measures the spread of the distribution.

这些积分在 X 的整个范围内计算。期望和方差保留其通常含义：均值表示中心位置，方差衡量分布的离散程度。

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6. The Continuous Uniform Distribution | 连续均匀分布

A continuous uniform distribution, denoted by X ~ U(a, b), describes a situation where any value in the interval [a, b] is equally likely. Its PDF is constant over that interval:

f(x) = 1/(b − a) for a ≤ x ≤ b, and 0 otherwise.

连续均匀分布，记作 X ~ U(a, b)，描述了区间 [a, b] 内任意值等可能的情况。其 PDF 在该区间上为常数：

f(x) = 1/(b − a)，a ≤ x ≤ b，否则为 0。

This is one of the simplest continuous distributions and is frequently used to model rounding errors or situations with no prior knowledge of a numerical quantity beyond its bounds.

这是最简单的连续分布之一，常用于对舍入误差或超出界限无先验知识的数值量进行建模。

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7. Mean and Variance of Uniform Distribution | 均匀分布的均值与方差

Using the expectation formulas, we can derive the mean and variance for X ~ U(a, b):

E(X) = (a + b)/2

Var(X) = (b − a)²/12

利用期望公式，我们可以推导出 X ~ U(a, b) 的均值与方差：

E(X) = (a + b)/2

Var(X) = (b − a)²/12

simple algebraic forms make the uniform distribution easy to handle in both theoretical and applied problems.

其 CDF 同样简单：对于 a ≤ x ≤ b，F(x) = (x − a)/(b − a)。这些简单的代数形式使得均匀分布在理论和应用问题中都易于处理。

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8. The Normal Distribution | 正态分布

The normal distribution, also known as the Gaussian distribution, is the most important continuous distribution in statistics. Its bell-shaped curve is symmetric about the mean μ, and its spread is determined by the standard deviation σ. The PDF for a normal random variable X ~ N(μ, σ²) is:

f(x) = ¹/_σ√2π e^{−(x−μ)²/(2σ²)}

正态分布，又称高斯分布，是统计学中最重要的连续分布。其钟形曲线关于均值 μ 对称，离散程度由标准差 σ 决定。正态随机变量 X ~ N(μ, σ²) 的 PDF 如下：

f(x) = ¹/_σ√2π e^{−(x−μ)²/(2σ²)}

Many natural phenomena—such as heights, weights, and measurement errors—follow an approximate normal distribution. Calculations often involve converting to the standard normal distribution.

许多自然现象——如身高、体重和测量误差——近似服从正态分布。计算通常涉及转化为标准正态分布。

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9. The Standard Normal Distribution | 标准正态分布

The standard normal distribution Z ~ N(0, 1) has mean 0 and variance 1. Any normal variable X can be standardised by subtracting μ and dividing by σ:

Z = (X − μ) / σ

标准正态分布 Z ~ N(0, 1) 的均值为 0，方差为 1。任何正态变量 X 都可以通过减去 μ 再除以 σ 进行标准化：

Z = (X − μ) / σ

The symmetry of the standard normal curve is especially useful: P(Z ≤ −z) = 1 − P(Z ≤ z).

标准化使我们能够使用单一的概率表。概率 P(X ≤ x) 等于 P(Z ≤ (x−μ)/σ)。标准正态曲线的对称性特别有用：P(Z ≤ −z) = 1 − P(Z ≤ z)。

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10. Using Statistical Tables | 使用统计表

en two values, subtract the table values: P(a < Z < b) = Φ(b) − Φ(a). For negative z-scores, use the property Φ(−z) = 1 − Φ(z). Always sketch a quick diagram to visualise the required area and avoid sign errors.

爱德思考试局提供标准正态累积分布表 Φ(z) = P(Z ≤ z)（正 z 值）。要计算 Z 落在两个值之间的概率，用表值相减：P(a < Z < b) = Φ(b) − Φ(a)。对于负 z 值，使用性质 Φ(−z) = 1 − Φ(z)。始终快速画图以直观显示所需面积，避免符号错误。

Typical questions require finding P(Z > c), P(Z < c), or a symmetric central interval. Working accurately with the table and performing inverse lookup (finding z given a probability) are essential exam skills.

常见题型包括求 P(Z > c)、P(Z < c) 或对称中心区间。准确使用表格并执行反向查找（给定概率求 z）是必要的考试技能。

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11. Finding Probabilities for General Normal | 求一般正态分布的概率

When X ~ rdise the boundary value x to z, then consult the standard normal table. For example, if X ~ N(100, 15²), to find P(X ≤ 115):

z = (115 − 100) / 15 = 1.00

当 X ~ N(μ, σ²) 时，首先将边界值 x 标准化为 z，然后查阅标准正态表。例如，X ~ N(100, 15²)，求 P(X ≤ 115)：

z = (115 − 100) / 15 = 1.00

he unknown is μ or σ, set up an equation using the given probability and solve using the inverse normal table. Such problems blend algebraic manipulation with statistical reasoning.

于是 P(X ≤ 115) = Φ(1.00) ≈ 0.8413。如果未知的是 μ 或 σ，则利用给定概率建立方程，并使用正态逆表求解。这类题目将代数操作与统计推理相结合。

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12. Summary and Key Tips | 总结与要点

Continuous distributions require a clear understanding of the link between PDF, CDF, integration, and differentiation. Bear in mind the following critical points:

连续分布需要清晰理解 PDF、CDF、积分和求导之间的联系。请牢记以下关键要点：

PDF f(x) – used to find probabilities via integration; total area = 1.	PDF f(x) – 用于通过积分求概率；总面积为 1。
CDF F(x) – gives P(X ≤ x); differentiate to get PDF.	CDF F(x) – 给出 P(X ≤ x)；求导可得 PDF。
Uniform distribution – constant PDF, linear CDF; quick mean and variance.	均匀分布 – PDF 为常数，CDF 为线性；均值方差计算快捷。
Normal distribution – bell-shaped; always standardise to Z before using tables.	正态分布 – 钟形；使用表格前务必标准化为 Z。
Symmetry of Z – often halves the work and helps with reverse lookups.	Z 的对称性 – 常使工作量减半，并有助于反向查找。

Regular practice with past papers and careful use of the formula booklet will strengthen your fluency. When sketching curves, mark the mean and standard deviation intervals to estimate probabilities visually before precise calculation. This habit reduces simple numerical mistakes and deepens your intuition for continuous data.

定期练习真题并仔细使用公式手册将提高你的熟练度。在绘制曲线时，标出均值和标准差区间，以便在精确计算前直观估计概率。这一习惯能减少简单的数字错误，并加深对连续数据的直觉。

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