📚 Continuous Random Variables | 连续随机变量
In Edexcel A Level Mathematics, continuous random variables model quantities that can take any value within an interval, such as time, length or temperature. Unlike discrete variables, probabilities are described by areas under a probability density function. This revision guide covers the key definitions, techniques and exam-style applications you need for the Statistics 2 paper.
在 Edexcel A Level 数学中，连续随机变量用于建模可在某一区间内取任意值的量，例如时间、长度或温度。与离散变量不同，连续变量的概率由概率密度函数下方的面积表示。本复习指南涵盖考试所需的核心定义、方法以及典型应用题。
1. What is a Continuous Random Variable? | 什么是连续随机变量
A continuous random variable X can take any value in a given interval, at least in theory. For example, the height of a randomly chosen student or the lifetime of a light bulb is continuous because it can be measured to any degree of accuracy. Other common examples include waiting times, weights, temperatures and distances.
连续随机变量 X 理论上可以取给定区间内的任意值。例如，随机选择的学生身高或灯泡的寿命都是连续的，因为它们可以被测量到任意精度。其他常见例子包括等待时间、重量、温度和距离。
Unlike a discrete random variable, which has countable outcomes such as 0, 1, 2, a continuous variable has uncountably many possible values. In practice, measurements are rounded, but the mathematical model treats the variable as continuous. This distinction changes how we calculate probabilities.
与离散随机变量不同，离散变量具有可数的结果，例如 0、1、2，而连续变量具有不可数多个可能值。实际测量中会四舍五入，但数学模型将变量视为连续的。这一区别改变了概率的计算方式。
In the Statistics 2 paper, you will often see continuous random variables defined on a finite interval [a, b] or on the whole real line. Always identify the domain first, because all integrals and probabilities depend on it.
在 Statistics 2 试卷中，连续随机变量常定义在有限区间 [a, b] 或整个实数轴上。始终先确定定义域，因为所有积分和概率都依赖于它。
2. Probability Density Function (PDF) | 概率密度函数
The behaviour of a continuous random variable X is described by a probability density function (PDF), usually written f(x). The PDF itself is not a probability; it is the height of the curve at x, and probability is represented by the area under f(x) between two values.
连续随机变量 X 的行为由概率密度函数（PDF）描述，通常记作 f(x)。PDF 本身不是概率；它是曲线在 x 处的高度，概率由 f(x) 下方两个值之间的面积表示。
For any interval [a, b], the probability that X lies between a and b is the area under the curve from a to b. This area is calculated using integration. The PDF can be greater than 1 for some values of x, because it is a density, not a probability.
对于任意区间 [a, b]，X 位于 a 与 b 之间的概率是曲线从 a 到 b 下方的面积。该面积通过积分计算。PDF 在某些 x 处可以大于 1，因为它是密度，不是概率。
The fundamental relationship is shown below. The total area under the PDF over all possible values must equal 1, because the total probability of all outcomes is 1.
基本关系如下所示。PDF 在所有可能值下方的总面积必须等于 1，因为所有结果的概率之和为 1。
P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx
∫ f(x) dx = 1 （从 −∞ 到 ∞）
3. Properties and Conditions of a Valid PDF | 有效 PDF 的条件
A function f(x) is a valid PDF for a continuous random variable over an interval if it satisfies two conditions. First, f(x) is non-negative everywhere in its domain. Second, the total area under f(x) over the whole domain equals 1.
若函数 f(x) 在某一区间上满足两个条件，则它可作为连续随机变量的有效 PDF。第一，f(x
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