Expectation and Variance of Sums of Independent Random Variables | 独立随机变量之和的期望与方差

📚 Expectation and Variance of Sums of Independent Random Variables | 独立随机变量之和的期望与方差

In probability theory and statistical inference, few results are as central as the expectation and variance of sums of random variables. When independent random variables are added, the expectation combines additively without any conditions, while the variance also admits a simple additive rule – provided independence holds. This article presents a rigorous yet accessible treatment of these theorems, tailored for IB Mathematics Analysis and Approaches HL students, with proofs, worked examples, and examination advice.

在概率论与统计推断中，很少有结果能像随机变量之和的期望与方差那样占据核心地位。当独立随机变量相加时，期望在任何条件下都满足可加性，而方差在独立性成立时同样具有简洁的加法法则。本文为IB数学分析与方法HL学生提供严谨而直观的推导，包含证明、实例与考试建议。

1. Linearity of Expectation | 期望的线性性质

The most fundamental result concerning sums of random variables is the linearity of expectation. For any two random variables X and Y, whether independent or not, the following identity always holds:

E(X + Y) = E(X) + E(Y)

This property extends naturally to any finite number of random variables X₁, X₂, …, Xₙ:

E(X₁ + X₂ + … + Xₙ) = E(X₁) + E(X₂) + … + E(Xₙ)

Notice that no independence assumption is needed. This is a remarkable feature of expectation: even strongly dependent variables satisfy this additive law. For example, if X represents the temperature and Y represents the humidity, the expected sum is simply the sum of the individual expectations, regardless of any meteorological relationship between them.

关于随机变量之和最基本的结果是期望的线性性质。对任意两个随机变量 X 和 Y，无论是否独立，以下恒等式总是成立：

E(X + Y) = E(X) + E(Y)

该性质天然地推广到任意有限个随机变量 X₁, X₂, …, Xₙ：

E(X₁ + X₂ + … + Xₙ) = E(X₁) + E(X₂) + … + E(Xₙ)

注意这里完全不需要独立性假设。这是期望的一大显著特点：即使变量之间高度相关，加法法则依然成立。例如，若 X 表示温度，Y 表示湿度，那么二者之和的期望就是各自期望之和，无论它们之间存在怎样的气象学关联。

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2. Proof of Linearity for Discrete Variables | 离散变量线性性质的证明

For discrete random variables, the proof uses the joint probability mass function. Let P(X = x, Y = y) denote the joint probability that X = x and Y = y. By definition:

E(X + Y) = Σₓ Σᵧ (x + y) · P(X = x, Y = y)

Splitting the sum into two parts gives:

E(X + Y) = Σₓ Σᵧ x·P(X = x, Y = y) + Σₓ Σᵧ y·P(X = x, Y = y)

In the first term, summing over all values of y produces the marginal probability P(X = x), since Σᵧ P(X = x, Y = y) = P(X = x). The first term therefore simplifies to Σₓ x·P(X = x) = E(X). By symmetry, the second term equals E(Y). This establishes the result without any reference to independence.

对于离散随机变量，证明需借助联合概率质量函数。设 P(X = x, Y = y) 表示 X = x 且 Y = y 的联合概率。根据定义：

E(X + Y) = Σₓ Σᵧ (x + y) · P(X = x, Y = y)

将求和拆分为两部分：

E(X + Y) = Σₓ Σᵧ x·P(X = x, Y = y) + Σₓ Σᵧ y·P(X = x, Y = y)

在第一项中，对所有 y 值求和即得边际概率 P(X = x)，因为 Σᵧ P(X = x, Y = y) = P(X = x)。因此第一项简化为 Σₓ x·P(X = x) = E(X)。同理，第二项等于 E(Y)。整个证明过程完全不涉及独立性。

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3. Review of Variance | 方差回顾

Variance measures the dispersion of a random variable around its mean. For a random variable X with mean μ = E(X), the variance is defined as:

Var(X) = E[(X – μ)²]

A computationally convenient equivalent form, obtained by expanding the square, is:

Var(X) = E(X²) – [E(X)]²

This alternative expression is particularly useful when calculating variances of sums, as we shall see shortly. The standard deviation, σ = √Var(X), is the square root of the variance and shares its units with the original variable.

方差衡量随机变量在其均值周围的离散程度。对于均值为 μ = E(X) 的随机变量 X，方差定义为：

Var(X) = E[(X – μ)²]

通过展开平方可得到计算上更方便的等价形式：

Var(X) = E(X²) – [E(X)]²

正如我们稍后将看到的，这一等价表达在计算和的方差时尤为有用。标准差 σ = √Var(X) 是方差的平方根，其单位与原始变量一致。

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4. Variance of a Sum of Independent Variables | 独立变量之和的方差

Unlike expectation, variance is not linear in general. The variance of a sum of two arbitrary random variables involves an additional term called the covariance:

Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y)

However, when X and Y are independent, the covariance is zero, and the variance addition rule emerges:

Var(X + Y) = Var(X) + Var(Y)

This is arguably the most important variance identity in the IB Mathematics syllabus. It is the foundation of the central limit theorem, the standard error formula, and countless inferential procedures. It is vital to recognise that independence is a sufficient condition for this rule, though not a necessary one – the rule holds whenever Cov(X, Y) = 0, which also occurs for uncorrelated (but possibly dependent) variables.

与期望不同，方差一般不具有线性性质。两个任意随机变量之和的方差还包含一个额外项——协方差：

Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y)

但当 X 和 Y 独立时，协方差为零，于是得到方差加法法则：

Var(X + Y) = Var(X) + Var(Y)

这可以说是IB数学教学大纲中最重要的方差恒等式。它是中心极限定理、标准误公式以及无数推断方法的基础。必须注意，独立性是该法则的充分条件而非必要条件——只要 Cov(X, Y) = 0 该法则就成立，而协方差为零也可能出现在不独立但互不相关的变量中。

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5. Proof of the Variance Addition Rule | 方差加法法则的证明

We now prove that if X and Y are independent, then Var(X + Y) = Var(X) + Var(Y). Starting from the computational formula:

Var(X + Y) = E[(X + Y)²] – [E(X + Y)]²

Expanding both terms:

Var(X + Y) = E(X² + 2XY + Y²) – [E(X) + E(Y)]²

= E(X²) + 2E(XY) + E(Y²) – [E(X)]² – 2E(X)E(Y) – [E(Y)]²

Since X and Y are independent, we have E(XY) = E(X)E(Y). The two cross terms 2E(XY) and -2E(X)E(Y) cancel exactly, leaving:

Var(X + Y) = E(X²) – [E(X)]² + E(Y²) – [E(Y)]²

Var(X + Y) = Var(X) + Var(Y)

The cancellation of the cross terms is the crux of the proof: independence ensures E(XY) = E(X)E(Y), which guarantees that the interaction between X and Y vanishes in the variance of the sum.

现在证明：若 X 和 Y 独立，则 Var(X + Y) = Var(X) + Var(Y)。从计算式出发：

Var(X + Y) = E[(X + Y)²] – [E(X + Y)]²

展开两项：

Var(X + Y) = E(X² + 2

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