连续随机变量II：期望方差全解 | CRV Part II: Expectation & Variance

📘 引言 / Introduction

在AQA A Level进阶数学统计学中，连续随机变量（Continuous Random Variables）是概率论的核心内容。掌握概率密度函数（PDF）、期望值、方差和标准差的计算方法，不仅对考试至关重要，也是理解高等统计学的基础。本篇基于AQA真题考点，系统梳理连续随机变量的期望与方差公式，帮助你在Paper中稳拿高分。

In AQA A Level Further Maths: Statistics, Continuous Random Variables (CRVs) form a core topic in probability theory. Mastering probability density functions (PDFs), expectation, variance, and standard deviation is essential not only for the exam but also for building a foundation in advanced statistics. This post, based on AQA past paper patterns, systematically reviews expectation and variance formulas for CRVs to help you secure top marks.

🔑 核心知识点 / Key Concepts

1. 连续随机变量的期望值 / Expectation of a CRV

连续随机变量 X 在区间 [a, b] 上的期望（均值）定义为：

E(X) = ∫ x·f(x) dx（积分区间从 a 到 b）

其中 f(x) 为概率密度函数（PDF）。这是所有可能取值的加权平均，权重由概率密度决定。计算时务必确认 f(x) 在定义域上的积分等于 1。

The expectation (mean) of a continuous random variable X defined on domain [a, b] is: E(X) = ∫ x·f(x) dx from a to b, where f(x) is the PDF. Always verify that the total area under f(x) equals 1 before proceeding.

2. 平方的期望与方差公式 / E(X²) and Variance

方差的快捷计算公式：Var(X) = E(X²) − [E(X)]²

其中 E(X²) = ∫ x²·f(x) dx。先分别计算 E(X) 和 E(X²)，再代入公式求方差，最后开平方得标准差 σ = √Var(X)。这是AQA考试最高频的计算路径。

The shortcut formula: Var(X) = E(X²) − [E(X)]², where E(X²) = ∫ x²·f(x) dx. Compute E(X) and E(X²) first, then subtract to get variance, and take the square root for standard deviation σ. This is the most frequently tested calculation pathway in AQA exams.

3. 线性变换的性质 / Properties of Linear Transformations

当 Y = aX + b 时：E(Y) = a·E(X) + b，Var(Y) = a²·Var(X)

注意：加常数 b 不影响方差，乘以常数 a 会使方差乘以 a²。标准差则乘以 |a|。这一性质在简化复杂随机变量的计算中非常实用。

For Y = aX + b: E(Y) = a·E(X) + b, Var(Y) = a²·Var(X). Adding a constant does not change the variance; multiplying by a scales variance by a². Standard deviation scales by |a|. This property is extremely useful for simplifying complex random variable calculations.

4. 非线性函数的期望 / Expectation of Non-Linear Functions

对于一般函数 g(X)：E[g(X)] = ∫ g(x)·f(x) dx

这一定义拓展了线性变换公式的适用范围。常见考题包括求 E(X³)、E(1/X) 等非线性变换的期望值。关键是代入正确的 g(x) 并在定义域上积分。

For a general function g(X): E[g(X)] = ∫ g(x)·f(x) dx. This generalizes beyond linear transformations. Common exam questions involve E(X³), E(1/X), etc. The key is substituting the correct g(x) and integrating over the defined domain.

5. 分段概率密度函数 / Piecewise PDFs

AQA真题中常出现分段定义的PDF。处理方法：将积分按定义域分成若干段，每段使用对应的 f(x) 表达式，分段计算后求和。分段点通常就是定义域的变化边界。

AQA past papers frequently feature piecewise-defined PDFs. Approach: split the integral at the domain boundaries, use the corresponding f(x) for each segment, calculate separately, then sum. The breakpoints are typically the domain boundaries where the PDF definition changes.

💡 学习建议 / Study Tips

• 熟记公式卡片：将 E(X)、E(X²)、Var(X)、线性变换四组公式做成记忆卡，考前反复过一遍。
• 分步计算不跳步：先求 E(X)，再求 E(X²)，最后求 Var(X)。每一步写出积分表达式，减少粗心错误。
• 验证PDF有效性：每次先检查 ∫ f(x) dx = 1，若不为 1 则题目可能有隐藏条件。
• 大量刷Past Papers：连续随机变量的题型规律性强，反复练习即可形成肌肉记忆。

• Memorize formula cards: Create flashcards for E(X), E(X²), Var(X), and linear transformation formulas — review before the exam.
• Step-by-step, no skipping: Compute E(X) → E(X²) → Var(X) in order. Write out the integral expressions at each step to avoid careless mistakes.
• Verify PDF validity: Always check ∫ f(x) dx = 1 first. If it doesn’t, there may be hidden conditions in the question.
• Practice past papers extensively: CRV questions follow predictable patterns — repeated practice builds muscle memory.

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