📚 Expected Value of a Discrete Random Variable | 离散随机变量的期望值
For Edexcel A-Level Mathematics, the expected value of a discrete random variable is one of the core ideas in statistics. It gives a single numerical summary of the centre of a probability distribution and underpins topics such as variance, linear transformations, binomial distribution, and hypothesis testing. This article explains what expected value means, how to calculate it, and how it is examined.
在 Edexcel A-Level 数学中，离散随机变量的期望值是统计学部分的核心概念之一。它用单一数值概括概率分布的中心，并为方差、线性变换、二项分布和假设检验等内容奠定基础。本文解释期望值的含义、计算方法以及考试中的常见考查方式。
1. What is a Discrete Random Variable? | 什么是离散随机变量？
A discrete random variable is a variable whose value depends on chance and can take only separate, countable values. Examples include the score on a fair six-sided die, the number of heads when tossing three coins, or the number of defective items in a sample of 10.
离散随机变量是一个取值依赖于随机性、且只能取有限个或可数个分离值的变量。例如掷一枚均匀六面骰子的得分、抛三枚硬币得到正面的次数、或 10 件样本中次品的数量。
In Edexcel exam questions, a random variable is usually written as an uppercase letter such as X, while a particular value it can take is written as a lowercase letter such as x. The expression P(X = x) means ‘the probability that the random variable X takes the value x’.
在 Edexcel 考试题中，随机变量通常用大写字母（如 X）表示，而它可以取的某个具体值用小写字母（如 x）表示。表达式 P(X = x) 表示“随机变量 X 取值为 x 的概率”。
2. Probability Distribution Recap | 概率分布回顾
Before calculating an expected value, you need a valid probability distribution. A probability distribution lists every possible value of X together with its probability. Two conditions must always be satisfied: each probability must be between 0 and 1 inclusive, and all probabilities must add up to exactly 1.
在计算期望值之前，你需要一个有效的概率分布。概率分布列出 X 的每个可能取值及其对应概率。必须始终满足两个条件：每个概率都介于 0 和 1 之间（含端点），并且所有概率之和恰好等于 1。
For example, suppose X is the number of heads when two fair coins are tossed. The possible values are 0, 1 and 2, with probabilities 0.25, 0.5 and 0.25. These probabilities sum to 1, so the distribution is valid.
例如，假设 X 是抛两枚均匀硬币时正面的次数。可能取值为 0、1、2，对应概率为 0.25、0.5、0.25。这些概率之和为 1，因此该分布有效。
| x | 0 | 1 | 2 |
|---|---|---|---|
| P(X = x) | 0.25 | 0.5 | 0.25 |
3. Definition of Expected Value E(X) | 期望值 E(X) 的定义
The expected value of a discrete random variable X, written E(X), is calculated by multiplying each possible value x by its probability P(X = x) and then summing all these products. It is defined by the formula below.
离散随机变量 X 的期望值记为 E(X)，计算方法是：将每个可能取值 x 与其对应概率 P(X = x) 相乘，然后将所有乘积相加。其定义由以下公式给出。
E(X) = Σ x P(X = x)
The summation is taken over all possible values of X. E(X) is also called the mean of X and is often denoted by the Greek letter μ.
求和范围覆盖 X 的所有可能取值。E(X) 也称为 X 的均值，通常用希腊字母 μ 表示。
4. Worked Example: Calculating E(X) | 计算 E(X) 的例题
Worked example: A discrete random variable X has the probability distribution shown below. Find E(X).
例题：离散随机变量 X 的概率分布如下表所示。求 E(X)。
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X = x) | 0.1 | 0.3 | 0.4 | 0.2 |
First check that the probabilities sum to 1: 0.1 + 0.3 + 0.4 + 0.2 = 1. Then multiply each value by its probability and add the results.
首先检查概率之和是否为 1：0.1 + 0.3 + 0.4 + 0.2 = 1。然后将每个取值与对应概率相乘，并把结果相加。
E(X) = 0 × 0.1 + 1 × 0.3 + 2 × 0.4 + 3 × 0.2
E(X) = 0 + 0.3 + 0.8 + 0.6 = 1.7
So the expected value of X is 1.7. Notice that 1.7 is not one of the actual possible values of X; this is an important point about expectation.
因此 X 的期望值为 1.7。注意 1.7 并不是 X 实际可能取到的值之一；这是期望值的一个重要特点。
5. Expected Value as a Long-Run Average | 期望值作为长期平均值
E(X) should be interpreted as a long-run average. If you repeated the underlying experiment many times and recorded the value of X each time, the average of those recorded values would get closer and closer to E(X). It is not the most likely value, and it does not have to be a possible value.
E(X) 应理解为长期平均值。如果你多次重复同一个随机试验，并每次记录 X 的取值，这些记录值的平均数会越来越接近 E(X)。它不一定是最可能出现的值，也不一定是随机变量的可能取值。
For the example above, E(X) = 1.7 means that over a very large number of trials, the average outcome per trial would be about 1.7. In a single trial, X can only be 0, 1, 2 or 3, but the average can lie between them.
就上述例子而言，E(X) = 1.7 表示在大量试验中，每次试验的平均结果约为 1.7。在单次试验中，X 只能取 0、1、2 或 3，但平均值可以落在这些值之间。
6. Linear Transformation: E(aX + b) | 线性变换 E(aX + b)
If a and b are constants, then the expected value of the linear transformation aX + b is given by the formula below. This rule is extremely useful when scores are rescaled, for example when a bonus is added or a score is doubled.
如果 a 和 b 是常数，那么线性变换 aX + b 的期望值由以下公式给出。当分数被重新调整时，例如加上奖励分或将分数翻倍，这个规则非常有用。
E(aX + b) = aE(X) + b
For instance, if a game pays 5 points for each success plus a fixed 3-point bonus, and X is the number of successes, then the score S = 5X + 3 has expected value E(S) = 5E(X) + 3. This follows directly from the definition because expectation is a linear operator.
例如，某个游戏每次成功得 5 分，外加固定的 3 分奖励，X 表示成功的次数，则得分 S = 5X + 3 的期望值为 E(S) = 5E(X) + 3。这一性质直接来自定义，因为期望是一个线性算子。
7. Variance and Expectation | 方差与期望
Expectation is also used to define variance. The variance of X is a measure of spread and is given by Var(X) = E((X − μ)²). In calculations, the practical formula below is usually easier to use.
期望值也用于定义方差。X 的方差是离散程度的度量，定义为 Var(X) = E((X − μ)²)。在实际计算中，以下实用公式通常更简便。
Var(X) = E(X²) − [E(X)]²
Here E(X²) is calculated using E(X²) = Σ x² P(X = x). For the worked distribution with x = 0, 1, 2, 3 and probabilities 0.1, 0.3, 0.4, 0.2, we get E(X²) = 0² × 0.1 + 1² × 0.3 + 2² × 0.4 + 3² × 0.2 = 0 + 0.3 + 1.6 + 1.8 = 3.7. Then Var(X) = 3.7 − 1.7² = 3.7 − 2.89 = 0.81.
其中 E(X²) 由公式 E(X²) = Σ x² P(X = x) 计算。对于上面 x = 0、1、2、3，概率为 0.1、0.3、0.4、0.2 的例题，可得 E(X²) = 0² × 0.1 + 1² × 0.3 + 2² × 0.4 + 3² × 0.2 = 0 + 0.3 + 1.6 + 1.8 = 3.7。因此 Var(X) = 3.7 − 1.7² = 3.7 − 2.89 = 0.81。
Many exam errors come from confusing E(X²) with [E(X)]². Always calculate E(X²) by squaring the x-values first, then taking the expectation.
许多考试错误都源于混淆 E(X²) 与 [E(X)]²。一定要先对 x 值平方，再求期望，从而计算 E(X²)。
8. Expected Value for Common Distributions | 常见分布的期望值
Certain named distributions have standard expected value formulas that Edexcel candidates are expected to know and use. The table below summarises the main ones.
一些常见命名分布有标准的期望值公式，Edexcel 考生需要掌握并会使用。下表总结了主要的几个分布。
| Distribution | Notation | Expected Value |
|---|---|---|
| Discrete uniform on 1, 2, …, n | X ~ U(1, n) | (n + 1) / 2 |
| Binomial | X ~ B(n, p) | np |
| Poisson | X ~ Po(λ) | λ |
For example, if X ~ B(20, 0.4), then E(X) = 20 × 0.4 = 8. If X ~ Po(3.5), then E(X) = 3.5. If X is the score on a fair six-sided die, then X is uniform on 1 to 6, so E(X) = (6 + 1) / 2 = 3.5.
例如，如果 X ~ B(20, 0.4)，则 E(X) = 20 × 0.4 = 8。如果 X ~ Po(3.5)，则 E(X) = 3.5。如果 X 表示一枚均匀六面骰子的得分，那么 X 服从 1 到 6 的均匀分布，因此 E(X) = (6 + 1) / 2 = 3.5。
9. Exam-Style Application: Fair Games and Decision-Making | 考试应用题：公平游戏与决策
Expected value is often tested in context. One common application is deciding whether a game is fair. If a player pays a cost C to play and receives a payoff X, the net gain is W = X − C. The game is fair when E(W) = 0, which means E(X) = C.
期望值经常在应用题中考查。一个常见应用是判断游戏是否公平。如果玩家支付成本 C 来玩游戏并获得收益 X，那么净收益为 W = X − C。当 E(W) = 0，即 E(X) = C 时，游戏是公平的。
For the earlier distribution where E(X) = 1.7, suppose the game costs 2 pounds to play. Then the net gain W = X − 2 has expected value E(W) = E(X) − 2 = 1.7 − 2 = −0.3. Over many games, the player would lose an average of 30 pence per game, so the game is not fair and favours the organiser.
对于前面 E(X) = 1.7 的分布，假设玩一次游戏需要 2 英镑。那么净收益 W = X − 2 的期望值为 E(W) = E(X) − 2 = 1.7 − 2 = −0.3。长期多次游戏时，玩家平均每局亏损 30 便士，因此该游戏不公平，对组织者有利。
If the game were instead priced at 1.7 pounds, then E(W) = 0 and the game would be fair. Expectation therefore provides a rational basis for comparing risky choices.
如果该游戏定价改为 1.7 英镑，则 E(W) = 0，游戏将是公平的。因此期望值为比较风险选择提供了理性依据。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
When answering Edexcel questions on expected value, always show each product x P(X = x) before summing. This earns method marks even if arithmetic slips. Check that all probabilities sum to 1 before starting; if they do not, you may need to find a missing probability first.
在回答 Edexcel 关于期望值的题目时，务必在求和前写出每个乘积 x P(X = x)。即使算术出错，也能获得方法分。计算之前先检查所有概率之和是否为 1；如果不是，可能需要先求出缺失的概率。
Common mistakes include using E(X²) = [E(X)]², forgetting that E(aX + b) changes by +b, and assuming E(X) must be a possible value. For variance of a linear transformation, remember that constants do not affect spread: Var(aX + b) = a² Var(X).
常见错误包括：误认为 E(X²) = [E(X)]²、忘记 E(aX + b) 需要加上常数 b、以及认为 E(X) 必须是随机变量的可能取值。对于线性变换的方差，要记住常数不影响离散程度：Var(aX + b) = a² Var(X)。
In exam solutions, define the random variable clearly, write the distribution table if one is not given, and state any standard formula you use such as E(X) = np for a binomial distribution. Clear working helps the examiner follow your reasoning and award marks consistently.
在考试解答中，要清晰地定义随机变量，如果没给出分布表就写出分布表，并写出你使用的标准公式，例如二项分布的 E(X) = np。清晰的解题过程可以帮助阅卷老师理解你的推理，从而更稳定地得分。
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