Continuous Random Variables | 连续随机变量

📚 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.

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。

For a continuous variable, P(X = a) = 0 for every single value a. This is a crucial difference from discrete variables and follows from the idea that a point has zero width, so the area above a single point is zero.

对于连续变量，对任意单个值 a 都有 P(X = a) = 0。这是与离散变量的关键区别，其原因是点没有宽度，因此单点上的面积为零。

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

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