Expectation of a Discrete Random Variable | 离散随机变量的期望值

📚 Expectation of a Discrete Random Variable | 离散随机变量的期望值

In many real-world problems we need a single number to summarise a random outcome. The expectation of a discrete random variable provides exactly this: a probability-weighted average of all possible values. It is used throughout statistics, from gambling games to financial models, and is a core topic in Edexcel A-Level Further Mathematics.

在许多实际问题中，我们需要用一个数字来概括随机结果。离散随机变量的期望值恰好提供了这一点：所有可能取值的概率加权平均。从赌博游戏到金融模型，期望值贯穿整个统计学，也是 Edexcel A-Level 进阶数学的核心内容。

1. Definition and Formula | 定义与公式

Let X be a discrete random variable with possible values x₁, x₂, …, xₙ and probability mass function f(x) = P(X = x). The expectation (or expected value) of X is the sum of each value multiplied by its probability:

设 X 为离散随机变量，可能取值为 x₁, x₂, …, xₙ，概率质量函数为 f(x) = P(X = x)。X 的期望值等于每个取值与其对应概率相乘后的总和：

E(X) = μ = Σ xᵢ f(xᵢ) = x₁f(x₁) + x₂f(x₂) + … + xₙf(xₙ)

Sometimes we write it more compactly as E(X) = Σ x f(x), where the summation runs over every possible value of X. Because the probabilities f(x) sum to 1, the expectation can be read as a weighted average of the values x, with the weights being the probabilities.

有时我们也写成更紧凑的形式：E(X) = Σ x f(x)，其中求和涵盖 X 的所有可能取值。由于概率 f(x) 的总和为 1，因此期望值可以理解为取值 x 以概率为权重的加权平均。

2. Using a Probability Table | 利用概率分布表计算

In A-Level questions you will usually be given a table showing each value of X and its corresponding probability. The expectation is then found by adding a row or column for xᵢf(xᵢ).

在 A-Level 考试中，通常会给出一个表格，列出 X 的每个取值及其概率。此时只需在表格中增加一行或一列

Published by TutorHao | A-Level 进阶数学 Revision Series | aleveler.com

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