📚 Expectation and Expected Value in IB Mathematics | 期望与期望值
Expectation, also called expected value, is a cornerstone concept in probability that captures the long-run average of a random variable. In IB Mathematics, mastering expectation allows you to analyse games of chance, evaluate financial decisions, and predict outcomes in scientific experiments. It turns uncertainty into a single, interpretable number. A random variable can take many values with many probabilities, yet expectation compresses all of that information into one summary that behaves predictably when experiments are repeated.
期望，也称期望值，是概率学的基石概念，它描述了随机变量在长期重复中的平均值。在 IB 数学中，掌握期望能帮助你分析机会游戏、评估金融决策并预测科学实验结果。它将不确定性转化为一个可解释的单一数值。一个随机变量可以取很多值、对应很多概率，而期望把这些信息压缩成一个摘要值，并且在实验反复进行时表现出稳定的规律。
1. What is Expectation? | 什么是期望？
For a random variable X, the expectation E(X) (or μ) is the probability-weighted average of all possible values X can take. Intuitively, if you were to repeat an experiment indefinitely, the sample mean would approach E(X). A classic example is a fair six-sided die: no single roll ever gives 3.5, yet 3.5 is exactly the expected value because each face has probability 1/6. Expectation therefore describes a long-run tendency rather than any individual outcome.
对于随机变量 X，期望 E(X)（或 μ）是 X 可能取的所有值的概率加权平均值。直观上，如果你无限次重复一个实验，样本均值将趋近 E(X)。一个经典例子是均匀六面骰子：没有任何一次投掷会给出 3.5，但 3.5 恰好是期望值，因为每个面出现的概率都是 1/6。因此期望描述的是长期趋势，而不是某一次具体结果。
Expectation condenses the entire distribution into a measure of central tendency. It is not necessarily one of the actual outcomes; rather, it locates the "balance point" of the probability mass. In the IB syllabus this idea appears whenever you are asked to find the mean of a discrete or continuous random variable, or to judge whether a game is fair.
期望将整个分布浓缩为一个集中趋势的度量。它不一定是实际可能发生的结果之一，而是概率质量的"平衡点"。在 IB 大纲中，只要题目要求你求离散型或连续型随机变量的均值，或者判断一个游戏是否公平，这个思想就会出场。
2. Notation and Formula for Discrete Random Variables | 离散随机变量的记号与公式
If X is a discrete random variable taking values x₁, x₂, x₃, …, xₙ with probabilities p₁, p₂, p₃, … re added. The probabilities must satisfy Σpᵢ = 1, otherwise the distribution is not valid.
如果 X 是离散型随机变量，取值为 x₁, x₂, x₃, …, xₙ，对应概率为 p₁, p₂, p₃, …, pₙ，那么期望由一个加权和定义：每个取值乘以该取值的概率，再把所有乘积相加。概率必须满足 Σpᵢ = 1，否则该分布不合法。
E(X) = Σ xᵢ pᵢ = Σ xᵢ P(X = xᵢ)
Two alternative symbols are widely used: E(X) emphasises the operation of "taking expectation", while μ emphasises that the result is the mean of the distribution. Probability tables in IB questions usually list the values in one row and the probabilities in another, and you simply compute the products in a third row before summing.
有两种符号被广泛使用：E(X) 强调"取期望"这个运算，而 μ 强调结果就是该分布的均值。IB 题目中的概率表通常把取值列在一行、概率列在另一行，你只需在第三行算出乘积，然后求和即可。
3. Worked Example: The Fair Six-Sided Die | 实例：均匀六面骰子
Let X be the score obtained when a fair six-sided die is rolled once. Each face has probability 1/6, so the expectation is the sum of each face multiplied by 1/6. The arithmetic gives 21/6, which simplifies to 3.5. This is a non-integer value that can never be rolled, which is precisely the point: expectation is a theoretical average, not a guaranteed outcome.
设 X 为投掷一次均匀六面骰子得到的点数。每个面概率为 1/6，因此期望是每个面乘以 1/6 之后的和。计算得到 21/6，化简为 3.5。这是一个不可能投出的非整数，而这正是关键：期望是理论平均值，不是必然出现的结果。
E(X) = (1 + 2 + 3 + 4 + 5 + 6) × (1/6) = 21/6 = 3.5
| xᵢ | 1 | 2 | 3 | 4 | 5 | 6 |
| P(X = xᵢ) | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |
| xᵢpᵢ | 1/6 | 2/6 | 3/6 | 4/6 | 5/6 | 6/6 |
1) + 3(0.1) + 4(0.1) + 5(0.1) + 6(0.5) = 4.5. The shift from 3.5 to 4.5 shows how sensitive expectation is to changes in the probability distribution.
如果骰子被做过手脚，使得 P(X = 6) = 1/2，而其他各面概率均为 1/10，则期望变为 4.5。计算为 1(0.1) + 2(0.1) + 3(0.1) + 4(0.1) + 5(0.1) + 6(0.5) = 4.5。从 3.5 到 4.5 的变化说明期望对概率分布的改变非常敏感。
4. Expectation as the Balance Point | 期望作为平衡点
A useful physical analogy treats the probability mass function as a set of weights placed on a ruler at positions x₁, x₂, …, xₙ. The expectation is exactly the point at which the ruler would balance. This is why μ is called a measure of location: it tells you where the centre of the distribution lies.
一个有用的物理类比是把概率质量函数看作放在直尺上 x₁, x₂, …, xₙ 位置的若干砝码。期望恰好就是这把尺子能够平衡的支点。这就是为什么 μ 被称为位置度量：它告诉你分布的中心在哪里。
This balance-point view also explains a frequent misconception. Because expectation is a centre of mass, it can lie outside the set of possible values, as with the die showing 3.5. If the distribution is skewed, the mean is pulled towards the long tail, which is why the mean, median and mode may differ.
这种平衡点的视角也解释了一个常见误解。正因为期望是质量中心，它可能落在所有可能取值之外，骰子的 3.5 就是如此。如果分布是偏斜的，均值会被长尾拉向一侧，这也是均值、中位数与众数可能不同的原因。
5. Expectation of a Function and Linear Transformations | 函数的期望与线性变换
If you apply a function to a random variable, the expectation of the new variable is not simply the function of the old expectation, except in special cases. However, expectation is always linear, which is the single most useful algebraic property in the whole topic.
如果你对随机变量施加某个函数，新变量的期望一般不等于旧期望的函数值，只有特殊情形才成立。但期望永远具有线性性质，这是整个主题中最有用的代数性质。
E(aX + b) = aE(X) + b
= c for any constant c, since a constant variable has only one possible value with probability 1. Linearity extends to sums: E(X + Y) = E(X) + E(Y) for any two random variables, whether or not they are independent.
这意味着把每个取值整体平移 b，期望也平移 b；把每个取值整体放大 a 倍，期望也放大 a 倍。由此还可得 E(c) = c，因为常数变量只有唯一取值，概率为 1。线性性质可以推广到和：对任意两个随机变量，E(X + Y) = E(X) + E(Y)，无论它们是否独立。
6. Variance and Standard Deviation Linked to Expectation | 与期望相关的方差与标准差
Expectation answers "where is the centre?", while variance answers "how spread out is the distribution?". Variance is defined as the expectation of the squared deviation from the mean, and it can be computed efficiently using the alternative formula.
期望回答"中心在哪里"，方差回答"分布有多分散"。方差定义为离均差平方的期望，并且可以用等价公式更高效地计算。
Var(X) = E((X − μ)²) = E(X²) − μ²
tandard deviation is the square root of the variance, so σ = √Var(X), and it carries the same units as X. For a linear transformation, Var(aX + b) = a²Var(X), which shows that shifting does not change spread.
使用这个捷径需要先算 E(X²) = Σ xᵢ²pᵢ，即平方值的概率加权平均，再减去均值的平方。标准差是方差的平方根，σ = √Var(X)，其单位与 X 相同。对线性变换有 Var(aX + b) = a²Var(X)，说明平移不改变分散程度。
7. Continuous Random Variables and the Integral Form | 连续随机变量与积分形式
When X is continuous with probability density function f(x), sums are replaced by integrals. The probability of any single point is zero, so expectation is built from the density over an interval.
当 X 是连续型随机变量、概率密度函数为 f(x) 时，求和被积分取代。任何单点的概率都是零，因此期望由区间上的密度积分构成。
E(X) = ∫ x f(x) dx
For a uniform distribution on the interval [a, b], the density is constant and the expectation is the midpoint (a + b)/2, which matches intuition. In IB questions the density is usually given as a simple polynomial on a closed interval, and the key steps are to confirm that the total area under f is 1 and then to integrate xf(x) over the support.
对于区间 [a, b] 上的均匀分布，密度为常数，期望就是中点 (a + b)/2，与直觉一致。在 IB 题目中，密度通常是在闭区间上给出的简单多项式，关键步骤是先确认 f 下方总面积为 1，再在支撑集上对 xf(x) 积分。
8. Combining Random Variables and Standard Distributions | 随机变量的组合与标准分布
nce this identity may fail. Several standard distributions have expectations worth memorising because they save time in the exam.
期望在加法下性质非常好，但乘法需要谨慎。如果 X 与 Y 独立，则 E(XY) = E(X)E(Y)；不独立时这个等式可能不成立。若干标准分布的期望值得记牢，因为能在考试中节省时间。
- Binomial distribution with n trials and probability p: E(X) = np.
- Poisson distribution with mean λ: E(X) = λ.
- Continuous uniform distribution on [a, b]: E(X) = (a + b)/2.
- Constant random variable: E(c) = c.
- 二项分布，n 次试验、成功概率 p：E(X) = np。
- 泊松分布，参数为 λ：E(X) = λ。
- 区间 [a, b] 上的连续均匀分布：E(X) = (a + b)/2。
- 常数随机变量：E(c) = c。
ndence, provided all expectations exist. This linearity is what makes expectation so convenient in multi-stage problems.
对于线性组合，E(aX + bY + c) = aE(X) + bE(Y) + c，无论变量是否相依，只要各期望存在即可。正是这种线性性使期望在多阶段问题中格外方便。
9. Applications: Games of Chance and Fair Prices | 应用：机会游戏与公平价格
(X) = 0 the game is fair; if E(X) is negative the player loses on average in the long run.
期望是判断游戏是否公平的天然工具。把 X 定义为玩家的净收益，计算 E(X) 并解释符号。若 E(X) = 0，游戏公平；若 E(X) 为负，则玩家在长期平均意义上亏损。
t gain is 0.5 − 1 = −0.5 per ticket. An organiser relying on thousands of tickets can expect a steady profit, even though individual buyers sometimes win big.
假设一张彩票售价 1 个单位，有千分之一的概率赢得 500 个单位。期望毛收益为 500 × (1/1000) = 0.5，因此每张票的期望净收益为 0.5 − 1 = −0.5。依靠成千上万张彩票，主办方可以获得稳定利润，尽管个别购买者偶尔会大赚。
10. Applications: Finance, Insurance and Risk | 应用：金融、保险与风险
In finance, expectation gives a baseline for comparing options with uncertain payoffs. An investor evaluating a project often starts with the expected return, then adjusts for risk, because two investments with the same expectation can have very different variances.
在金融中，期望为比较收益不确定的方案提供了基准。投资者评估项目时通常先看期望收益，再根据风险进行调整，因为两个期望相同的投资，其方差可能差别很大。
.02 × 50,000 = 1,000. The premium must exceed this figure to cover administrative costs and profit, which explains why the insured person typically has a negative expected monetary gain but buys protection against a catastrophic loss.
保险定价是其直接应用。如果事故发生概率为 0.02、赔付额为 50,000，那么每份保单的期望赔付为 0.02 × 50,000 = 1,000。保费必须高于这个数字才能覆盖管理成本与利润，这也解释了为什么投保人的期望货币收益通常为负，却仍愿意购买保障以规避灾难性损失。
11. Applications: Scientific Experiments and Decision Making | 应用：科学实验与决策
In experimental science, expectation underpins the practice of repeating measurements and taking the mean. Random errors tend to cancel, so the sample mean converges towards the true expected value as the number of trials grows, which is the essence of the law of large numbers.
在实验科学中，期望支撑了"重复测量取平均"的做法。随机误差倾向于相互抵消，因此随着试验次数增加，样本均值会趋近真实的期望值，这正是大数定律的核心。
Expectation also supports decision making under uncertainty, for example choosing between two treatments with different success probabilities and payoffs, or deciding how many items to stock when demand is random. In each case the expected value provides a defensible benchmark, although risk-averse decision makers may also look at the variance or at worst-case outcomes.
期望也支持不确定性下的决策，例如在两种成功率与收益不同的治疗方案之间选择，或在需求随机时决定库存数量。在这些情形中，期望值提供了一个有依据的基准，不过厌恶风险的决策者还会关注方差或最坏情形。
12. Common Mistakes and Exam Technique | 常见错误与考试技巧
The most frequent errors in expectation questions are mechanical rather than conceptual. Checking the probability sum, keeping track of squares, and interpreting the final number in context will protect most of your marks.
期望题目中最常见的错误往往是操作层面的，而非概念层面的。检查概率之和、留意平方项、并结合情境解释最终数值，就能守住大部分分数。
- Forgetting to verify that the probabilities sum to 1 before computing E(X).
- Confusing E(X²) with [E(X)]²; the two are equal only when the variance is zero.
- Assuming E(X) must be one of the possible values of X.
- Applying E(XY) = E(X)E(Y) without checking independence.
- Omitting units or failing to state whether a game is fair, favourable or unfavourable.
- Rounding intermediate products too early, which distorts the final answer.
- 在计算 E(X) 之前忘记验证概率之和是否为 1。
- 把 E(X²) 与 [E(X)]² 混为一谈；只有当方差为零时二者才相等。
- 误以为 E(X) 必须是 X 的某个可能取值。
- 未检查独立性就使用 E(XY) = E(X)E(Y)。
- 漏写单位，或没有说明游戏是公平、有利还是不利。
- 过早对中间乘积四舍五入，导致最终答案失真。
text of the question. When variance is also required, compute E(X²) in the same table and finish with Var(X) = E(X²) − μ².
可靠的考试流程是：列出取值与概率，检验 Σpᵢ = 1，计算 Σxᵢpᵢ，然后用一句话结合题目情境解释结果。如果还需要方差，就在同一张表中算出 E(X²)，最后用 Var(X) = E(X²) − μ² 收尾。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图