Probability Density Functions and Combining Random Variables | 概率密度函数与随机变量组合

📚 Probability Density Functions and Combining Random Variables | 概率密度函数与随机变量组合

In Edexcel A-Level Mathematics, particularly in Statistics 2 (S2) and Further Mathematics, mastering probability density functions (PDFs) and the rules for combining independent random variables is essential. A continuous random variable can take any value in an interval, and its distribution is modelled by a PDF. This article will guide you through the key concepts — from verifying valid PDFs, calculating probabilities and cumulative distributions, to finding expectations, medians, and modes. We then extend these ideas to linear combinations of independent variables and sums of normal distributions. Through a consistent worked example, you will see how all these topics connect, preparing you for typical exam questions.

在爱德思A-Level数学中，特别是在统计2（S2）和进阶数学中，掌握概率密度函数（PDF）以及独立随机变量的组合规则至关重要。连续随机变量可以取某一区间内的任意值，并通过PDF来描述其分布。本文将带你梳理核心概念——从验证有效的PDF、计算概率和累积分布，到求解期望、中位数和众数。随后我们将思路延伸到独立变量的线性组合与正态分布之和。通过一个贯穿始终的例题，你将看到这些知识点如何关联起来，从而为应对典型考题做好准备。

1. Continuous Random Variables | 连续随机变量

A continuous random variable can take infinitely many values within a given range. Instead of assigning probability to individual points, we define probabilities over intervals using a probability density function, f(x). For a continuous variable X, the probability that X equals any exact value is zero; only intervals have non-zero probabilities. The entire area under the graph of f(x) must equal 1, representing the total probability.

连续随机变量可以在给定的范围内取无限多个值。我们不能对个别点赋予概率，而是通过概率密度函数 f(x) 对区间上的概率加以定义。对于连续变量 X，X 恰好取某一具体值的概率为零；只有区间才具有非零的概率。f(x) 曲线下的全部面积必须等于1，代表总概率。

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

A PDF, written as f(x), is a function used to describe the likelihood of a continuous random variable falling near a given value. To obtain the probability that X lies between a and b, we integrate the PDF: P(a ≤ X ≤ b) = ∫_a^b f(x) dx. As a working example, consider f(x) = kx² for 0 ≤ x ≤ 3. To determine the constant k, we set the total area to 1: ∫₀³ kx² dx = 1. Integrating gives k[(1/3)x³]₀³ = k × 9 = 1, so k = 1/9. The PDF is therefore f(x) = (1/9)x², 0 ≤ x ≤ 3. This PDF will be used throughout many of the following sections to illustrate concepts.

PDF 写作 f(x)，是用于描述连续随机变量在某一数值附近出现的可能性的函数。要得到 X 落在 a 和 b 之间的概率，需对 PDF 积分：P(a ≤ X ≤ b) = ∫_a^b f(x) dx。我们以一个例题贯穿讲解：考虑 f(x) = kx²，0 ≤ x ≤ 3。为确定常数 k，令总面积等于1：∫_{Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com}

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