📚 A - Continuous Random Variables | A - 连续随机变量
In IB Mathematics, the study of continuous random variables extends the ideas of probability distributions to outcomes that can take any value within an interval on the real number line. Unlike discrete random variables, which have probability mass at specific points, continuous random variables are described by probability density functions (PDFs) and cumulative distribution functions (CDFs). Mastering this topic involves understanding how to calculate probabilities, the median, percentiles, expectation, variance, and applying linear transformations. The normal distribution plays a central role as a model for many natural phenomena. This article provides a comprehensive walkthrough of continuous random variables as required for the IB syllabus, with clear explanations and examples.
在IB数学中，连续随机变量的学习将概率分布的概念扩展到可以在实数轴区间内取任何值的结果。与在特定点上具有概率质量的离散随机变量不同，连续随机变量通过概率密度函数（PDF）和累积分布函数（CDF）来描述。掌握这个主题需要理解如何计算概率、中位数、百分位数、期望、方差，并应用线性变换。正态分布作为许多自然现象的模型，扮演核心角色。本文提供了IB大纲所要求的连续随机变量的全面讲解，包括清晰的解释和例子。
1. What is a Continuous Random Variable? | 什么是连续随机变量？
A continuous random variable is a variable that can take any value within a given range or interval. For example, the height of a student, the time taken to complete a task, or the mass of an apple are all continuous measurements. Since the set of possible outcomes is uncountably infinite, we cannot assign a probability to each individual value. Instead, probabilities are determined over intervals.
连续随机变量是指在一个给定范围或区间内可以取任何值的变量。例如，学生的身高、完成任务所需的时间或一个苹果的质量都是连续测量值。由于可能结果的集合是不可数的无限集，我们无法为每个单独的值分配概率。相反，概率是在区间上确定的。
a fundamental difference from discrete random variables. Therefore, when working with continuous distributions, we only consider probabilities of intervals, such as P(a < X < b).
对于任何连续随机变量 X，X 等于任何一个精确值 c 的概率为零：P(X = c) = 0。这是与离散随机变量的根本区别。因此，在处理连续分布时，我们只考虑区间的概率，例如 P(a < X < b)。
2. Probability Density Function (PDF) | 概率密度函数
The probability density function, denoted f(x), describes the relative likelihood of a continuous random variable taking on a given value. Unlike discrete probability mass functions, f(x) itself does not give a probability directly. The area under the curve of f(x) over an interval gives the probability that the variable falls within that interval.
概率密度函数，记作 f(x)，描述了连续随机变量取某个值的相对可能性。与离散概率质量函数不同，f(x) 本身并不直接给出概率。f(x) 曲线下某个区间上的面积给出了变量落在这个区间内的概率
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