Expectation Algebra | 期望代数

📚 Expectation Algebra | 期望代数

Expectation algebra provides a powerful set of tools for manipulating expected values, variances, and covariances without performing lengthy summations or integrations each time. In IB Mathematics, a solid grasp of these properties allows you to solve probability and statistics problems with speed and accuracy, whether dealing with discrete or continuous random variables.

期望代数提供了一套强大的工具，使我们无需每次都进行冗长的求和或积分即可操作期望值、方差和协方差。在 IB 数学中，熟练掌握这些性质能让你快速准确地解决概率与统计问题，无论面对的是离散还是连续随机变量。

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1. Definition of Expectation | 期望的定义

The expected value E(X), often called the mean, is the probability-weighted average of all possible values a random variable X can take. It represents the long-run average outcome of a random process.

期望值 E(X)，通常称为均值，是随机变量 X 所有可能取值的概率加权平均。它代表了随机过程的长期平均结果。

For a discrete random variable with values x₁, x₂, …, xₙ and corresponding probabilities p₁, p₂, … andom variable with probability density function f(x), E(X) = ∫ x f(x) dx over the domain.

对于离散随机变量，设其取值为 x₁, x₂, …, xₙ，对应概率为 p₁, p₂, …, pₙ，则期望为 E(X) = Σ xᵢ pᵢ。对于具有概率密度函数 f(x) 的连续随机变量，E(X) = ∫ x f(x) dx 在定义域上积分。

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2. Linearity of Expectation | 期望的线性性质

Y) = aE(X) + bE(Y). This holds regardless of whether X and Y are independent or dependent.

最重要的性质之一是线性性：对于任意随机变量 X 和 Y 以及常数 a 和 b，恒有 E(aX + bY) = aE(X) + bE(Y)。无论 X 和 Y 是否独立，该性质始终成立。

This result extends to any finite number of random variables: E(c₁X₁ + c₂X₂ + … + cₙXₙ) = c₁E(X₁) + c₂E(X₂) + … ), where cᵢ are constants. The proof relies on the linearity of summation and integration.

这一结果可推广到任意有限个随机变量：E(c₁X₁ + c₂X₂ + … + cₙXₙ) = c₁E(X₁) + c₂E(X₂) + … + cₙE(Xₙ)，其中 cᵢ 是常数。其证明依赖于求和与积分的线性性质。

A direct consequence is that expectation can be "distributed" across addition and subtraction: E(X + Y - Z) = E(X) + E(Y) - E(Z).

直接推论是，期望可以“分配”到加减运算中：E(X + Y - Z) = E(X) + E(Y) - E(Z)。

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3. Expectation of Constants and Scaling | 常数与缩放的期望

c) = c. This is because a constant has no variability; its average is the constant itself.

对于任意常数 c，c 的期望值就是 c 本身：E(c) = c。这是因为常数没有变异性，其平均值就是该常数。

es linearly: E(aX) = aE(X). This follows from taking b = 0 or Y = 0 in the linearity rule.

如果将一个随机变量乘以常数 a，期望会按比例改变：E(aX) = aE(X)。这可以从线性性质中令 b = 0 或 Y = 0 得出。

X) + c. Combining both, we obtain the general linear function rule: E(aX + b) = aE(X) + b.

类似地，加上一个常数会使期望平移该常数：E(X + c) = E(X) + c。将两者结合，我们得到一般线性函数的期望公式：E(aX + b) = aE(X) + b。

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4. Expectation of a Linear Function in Practice | 线性函数的期望应用

b we directly obtain E(W) = 3×5 - 7 = 8. No further distributional information is needed.

假设 X 是一个随机变量，E(X) = 5。如果我们构造一个新变量 W = 3X - 7，那么使用 E(aX + b) = aE(X) + b 可直接得到 E(W) = 3×5 - 7 = 8，无需了解分布的其他信息。

This illustrates why expectation algebra is so efficient: once you know E(X), you can instantly find the expectation of any linear transformation. In IB exam questions, you will often be given E(X) and Var(X) and asked to compute the mean of transformed variables.

这说明期望代数为何如此高效：一旦知道 E(X)，就能立刻求出任意线性变换的期望。在 IB 考试中，经常会给出 E(X) 和 Var(X)，要求计算变换后的变量的均值。

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5. Definition of Variance | 方差的定义

ned as the expected squared deviation from the mean: Var(X) = E[(X - μ)²], where μ = E(X).

随机变量 X 的方差衡量其分布的离散程度。它定义为与均值偏差平方的期望：Var(X) = E[(X - μ)²]，其中 μ = E(X)。

= E(X²) - [E(X)]². This comes from expanding (X - μ)² and using linearity of expectation.

一个等价且在计算上通常更方便的公式是 Var(X) = E(X²) - [E(X)]²。这可以通过展开 (X - μ)² 并运用期望的线性性质得出。

X) is always non-negative, and Var(X) = 0 if and only if X is constant with probability 1.

在进行代数运算时，请记住 Var(X) 总是非负的，且 Var(X) = 0 当且仅当 X 以概率 1 为常数。

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6. Variance of a Linear Transformation | 线性变换的方差

hat the constant b has no effect on variance, since adding a constant simply shifts the distribution without changing its spread.

与期望的一个关键区别是，方差不具有线性性。对于线性函数 aX + b（其中 a 和 b 为常数），方差变换为 Var(aX + b) = a² Var(X)。注意常数 b 对方差没有影响，因为加上常数只是平移分布而不改变其离散程度。

E[(aX + b - E(aX + b))²] = E[(a(X - μ))²] = a² E[(X - μ)²] = a² Var(X). The b cancels out.

这可以通过定义证明：Var(aX + b) = E[(aX + b - E(aX + b))²] = E[(a(X - μ))²] = a² E[(X - μ)²] = a² Var(X)。b 被抵消了。

Var(X) + b or to forget the square on a. Always square the multiplier: Var(3X) = 9 Var(X).

一个常见错误是写成 Var(aX + b) = a² Var(X) + b 或忘记对 a 取平方。一定要将乘数平方：Var(3X) = 9 Var(X)。

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7. Expectation of Products and Independence | 乘积的期望与独立性

independence can be assumed, we have E(XY) = E(X)E(Y), which is a powerful simplification.

一般来说，E(XY) 并不简单地等于 E(X)E(Y)。只有当 X 和 Y 为独立随机变量时，这个等式才成立。如果可以假设独立性，我们就有 E(XY) = E(X)E(Y)，这是一个非常强大的简化。

tion using summation or double integration, or by applying the definition E(XY) = Σ Σ xᵢ yⱼ pᵢⱼ. The difference between E(XY) and E(X)E(Y) is captured by covariance.

对于不独立的变量，E(XY) 必须利用联合分布通过求和或二重积分计算，或应用定义 E(XY) = Σ Σ xᵢ yⱼ pᵢⱼ。E(XY) 与 E(X)E(Y) 之间的差异由协方差来体现。

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8. Covariance and Its Properties | 协方差及其性质

E(Y))]. It measures the direction and strength of the linear relationship between X and Y.

两个随机变量 X 和 Y 的协方差定义为 Cov(X,Y) = E[(X - E(X))(Y - E(Y))]。它衡量 X 和 Y 之间线性关系的方向和强度。

Y) - E(X)E(Y). This follows from expanding the product and using linearity of expectation.

一个更方便的计算公式为 Cov(X,Y) = E(XY) - E(X)E(Y)。这可以通过展开乘积并运用期望的线性性质得到。

X,Y). The additive constants b and d disappear, and the multiplicative constants factor out.

协方差满足若干代数规则：Cov(aX + b, cY + d) = ac Cov(X,Y)。加性常数 b 和 d 会消失，乘性常数会被提取出来。

owever, the converse is not necessarily true: zero covariance does not imply independence.

此外，Cov(X,X) = Var(X)，且协方差是对称的：Cov(X,Y) = Cov(Y,X)。如果 X 和 Y 独立，则 Cov(X,Y) = 0。但反过来不一定成立：零协方差不意味着独立。

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9. Variance of Sums and Differences | 和与差的方差

ov(X,Y). This is obtained by expanding Var(X + Y) = E[(X + Y - μₓ - μᵧ)²] and simplifying.

对于任意两个随机变量 X 和 Y，其和的方差为 Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y)。这可以通过展开 Var(X + Y) = E[(X + Y - μₓ - μᵧ)²] 并化简得到。

- 2Cov(X,Y). Notice it is plus the individual variances but minus twice the covariance, because subtracting Y effectively reverses its sign in the interaction term.

差的方差非常相似：Var(X - Y) = Var(X) + Var(Y) - 2Cov(X,Y)。注意它是个体方差之和，但减去两倍的协方差，这是因为减去 Y 实际上反转了它在交互项中的符号。

ar(X ± Y) = Var(X) + Var(Y). This result extends to n independent variables: Var(X₁ + X₂ + … + Xₙ) = Var(X₁) + Var(X₂) + … + Var(Xₙ).

当 X 和 Y 独立时，Cov(X,Y) = 0，因此两个公式都简化为 Var(X ± Y) = Var(X) + Var(Y)。这一结果可推广至 n 个独立变量：Var(X₁ + X₂ + … + Xₙ) = Var(X₁) + Var(X₂) + … + Var(Xₙ)。

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10. Combining Linear Combinations | 线性组合的综合

bY + c, where X and Y are random variables and a, b, c are constants. By combining the rules, we get:

考虑线性组合 W = aX + bY + c，其中 X 和 Y 是随机变量，a, b, c 为常数。综合各项规则，我们得到：

E(W) = aE(X) + bE(Y) + c

Var(W) = a² Var(X) + b² Var(Y) + 2ab Cov(X,Y)

la you will frequently apply in IB problems involving sums of independent normal variables or in portfolio analysis.

如果 X 和 Y 独立，协方差项为零，则 Var(W) = a² Var(X) + b² Var(Y)。这是在 IB 问题中经常应用的公式，例如涉及独立正态变量之和或投资组合分析。

To remember the pattern: expectation combines additively; variance combines with squared coefficients plus twice the covariance for each pair. This mirrors expanding (aX + bY)² and taking expectations.

记忆模式：期望按加性组合；方差按系数平方相加，再为每对变量加上两倍协方差。这类似于展开 (aX + bY)² 并取期望。

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11. Common Mistakes and Misconceptions | 常见错误与误区

In fact, E(X²) is generally larger than [E(X)]² unless X is constant. The difference is Var(X).

错误一：假设 E(X²) = [E(X)]²。事实上，E(X²) 通常大于 [E(X)]²，除非 X 是常数。两者之差正是 Var(X)。

and mistakenly including b in the variance, e.g. writing a² Var(X) + b² or a² Var(X) + b.

错误二：忘记 Var(aX + b) = a² Var(X)，错误地将 b 包含在方差中，例如写成 a² Var(X) + b² 或 a² Var(X) + b。

es do not hold in general; expectations do not distribute over division or non-linear functions.

错误三：假设 E(X/Y) = E(X)/E(Y) 或 E(1/X) = 1/E(X)。这些等式一般不成立；期望不能对除法或非线性函数进行分配。

iance only means there is no linear relationship, but a non-linear dependence could still exist.

错误四：仅凭 Cov(X,Y) = 0 就假设变量独立。零协方差只意味着不存在线性关系，但非线性的依赖关系仍然可能存在。

without checking independence. Always verify that the problem states independence or that it can be safely deduced from the context.

错误五：未检查独立性就应用 E(XY) = E(X)E(Y)。务必确认题目中声明了独立性，或能从上下文合理推断。

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12. Worked Example and Quick Summary Table | 例题与速查表

with E(X) = 4, Var(X) = 2, E(Y) = 1, Var(Y) = 3. Find E(2X - 3Y + 5) and Var(2X - 3Y + 5).

例题：设 X 和 Y 相互独立，E(X) = 4，Var(X) = 2，E(Y) = 1，Var(Y) = 3。求 E(2X - 3Y + 5) 和 Var(2X - 3Y + 5)。

Using expectation linearity: E(2X - 3Y + 5) = 2E(X) - 3E(Y) + 5 = 2×4 - 3×1 + 5 = 10.

利用期望的线性性：E(2X - 3Y + 5) = 2E(X) - 3E(Y) + 5 = 2×4 - 3×1 + 5 = 10。

Cov(X,Y) = 0. Then Var(2X - 3Y + 5) = 2² Var(X) + (-3)² Var(Y) = 4×2 + 9×3 = 8 + 27 = 35.

由于 X 和 Y 独立，Cov(X,Y) = 0。因此 Var(2X - 3Y + 5) = 2² Var(X) + (-3)² Var(Y) = 4×2 + 9×3 = 8 + 27 = 35。

The following table summarises the key expectation algebra rules:

下表总结了关键的期望代数规则：

• E(c) = c
• E(aX + b) = aE(X) + b
• E(X ± Y) = E(X) ± E(Y)
• E(XY) = E(X)E(Y) [if independent]
• Var(X) = E(X²) - [E(X)]²
• Var(aX + b) = a² Var(X)
• Var(X ± Y) = Var(X) + Var(Y) ± 2Cov(X,Y)
• Var(X ± Y) = Var(X) + Var(Y) [if independent]
• Cov(X,Y) = E(XY) - E(X)E(Y)
• Cov(aX + b, cY + d) = ac Cov(X,Y)

Memorising these identities and understanding when each applies will give you a strong foundation for tackling probability questions in the IB exam. Practice by first identifying which variables are independent and then carefully applying the correct variance formula with squared coefficients.

熟记这些恒等式并理解各自的适用条件，将为应对 IB 考试中的概率题打下坚实基础。练习时，先识别哪些变量是独立的，然后仔细选用正确的方差公式并对方差的系数取平方。

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