📚 Combining Random Variables | 随机变量的组合
When working with probability and statistics, we often need to analyse new random variables formed by adding, subtracting, or scaling existing ones. Understanding how expectations, variances, and distributions combine is essential for modelling real-world phenomena, from investment portfolios to measurement errors. This article explores the rules for combining random variables, the role of independence, and applications to the normal and other common distributions, closely aligned with the IB Mathematics curriculum.
在处理概率与统计时，我们经常需要分析通过加减或缩放已有随机变量而形成的新变量。理解期望、方差和分布如何组合，对于从投资组合到测量误差等现实世界的建模至关重要。本文探讨组合随机变量的规则、独立性的作用，以及正态分布和其他常见分布的应用，紧扣IB数学大纲。
1. Introduction to Random Variable Combinations | 随机变量组合简介
A linear combination of random variables X and Y takes the form aX + bY, where a and b are constants. Such combinations arise naturally: total profit from two products, difference in test scores, or a weighted portfolio of assets. The key is to determine the distribution, expected value, and variability of the new variable.
随机变量 X 和 Y 的线性组合形式为 aX + bY，其中 a 和 b 为常数。这种组合自然地出现：两种产品的总利润、考试分数的差值或资产的加权组合。关键在于确定新变量的分布、期望值和变异性。
2. Expectation of Linear Combinations | 线性组合的期望
s: E(∑aᵢXᵢ) = ∑aᵢE(Xᵢ). This property underpins portfolio returns and unbiased estimators.
线性组合的期望总是线性的，与独立性无关：E(aX + bY) = aE(X) + bE(Y)。这源自积分或求和的线性性质。它可推广到任意数量的变量：E(∑aᵢXᵢ) = ∑aᵢE(Xᵢ)。这一性质支撑着投资组合回报和无偏估计量。
3. Variance of Linear Combinations: Independent Case | 独立情况下线性组合的方差
− bY) = a²Var(X) + b²Var(Y) when independent, as subtraction does not reduce variability.
当 X 和 Y 独立时，方差可优美地简化为：Var(aX + bY) = a²Var(X) + b²Var(Y)。协方差项消失，因为 Cov(X, Y) = 0。注意常数被平方，这反映了缩放对分散程度的影响。对于差值 aX − bY，公式保持不变：独立时，Var(aX − bY) = a²Var(X) + b²Var(Y)，因为减法并不会减少变异性。
4. Covariance and Non-independent Variables | 协方差与非独立变量
uce it. For aX − bY, the sign changes: Var(aX − bY) = a²Var(X) + b²Var(Y) − 2ab Cov(X, Y).
如果 X 和 Y 不独立，则必须包含协方差：Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab Cov(X, Y)。协方差衡量联合变异性。正协方差表示它们倾向同向变动，增加组合方差；负协方差则可降低组合方差。对于 aX − bY，符号改变：Var(aX − bY) = a²Var(X) + b²Var(Y) − 2ab Cov(X, Y)。
5. Combining Independent Normal Random Variables | 组合独立正态随机变量
If X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²) are independent, any linear combination aX + bY also follows a normal distribution: aX + bY ~ N(aμ₁ + bμ₂, a²σ₁² + b²σ₂²). This reproductive property makes the normal distribution extremely useful in inference. The sum of independent normals is normal; even the difference remains normal with variance equal to the sum of variances.
若 X ~ N(μ₁, σ₁²) 与 Y ~ N(μ₂, σ₂²) 独立，则任意线性组合 aX + bY 也服从正态分布：aX + bY ~ N(aμ₁ + bμ₂, a²σ₁² + b²σ₂²)。这一再生性质使得正态分布在推断中极其有用。独立正态变量之和为正态；甚至差值也为正态，其方差等于方差之和。
6. Distribution of Sums of Discrete Random Variables | 离散随机变量之和的分布
For discrete distributions, sums of independent variables often produce recognisable distributions. If X ~ B(n₁, p) and Y ~ B(n₂, p) are independent binomials with the same success probability, then X + Y ~ B(n₁ + n₂, p). Similarly, if X ~ Po(λ₁) and Y ~ Po(λ₂) are independent Poisson variables, X + Y ~ Po(λ₁ + λ₂). These additive properties require identical parameters (except the index for binomial) and independence.
对于离散分布，独立变量之和常产生可识别的分布。若 X ~ B(n₁, p) 和 Y ~ B(n₂, p) 是独立且成功概率相同的二项分布，则 X + Y ~ B(n₁ + n₂, p)。类似地，若 X ~ Po(λ₁) 与 Y ~ Po(λ₂) 为独立泊松变量，则 X + Y ~ Po(λ₁ + λ₂)。这些可加性要求参数一致（除二项分布的次数外）且独立。
7. Combining Means of Random Samples | 组合随机样本的均值
When we take a random sample X₁, X₂, …, Xₙ from a population with mean μ and variance σ², the sample mean X̄ = (∑Xᵢ)/n is itself a random variable. Its expected value is E(X̄) = μ, and if the Xᵢ are independent, Var(X̄ m two populations, follows the same linear combination rules for expectation and variance.
当我们从均值为 μ、方差为 σ² 的总体中抽取随机样本 X₁, X₂, …, Xₙ 时，样本均值 X̄ = (∑Xᵢ)/n 本身也是一个随机变量。其期望值为 E(X̄) = μ，且若 Xᵢ 独立，则 Var(X̄) = σ²/n。这表明平均可降低变异性。组合两个独立样本均值（例如来自两个总体）时，期望和方差同样遵循线性组合规则。
8. Difference of Two Sample Means (Independent) | 两个样本均值之差（独立）
Consider two independent samples: one of size n₁ from N(μ₁, σ₁²), the other of size n₂ from N(μ₂, σ₂²). The difference X̄₁ − X̄₂ is normal with mean μ₁ − μ₂ and variance (σ₁²/n₁) + (σ₂²/n₂). Even if the populations are not normal, by the Central Limit Theorem the distribution of the difference is approximately normal for large samples, enabling confidence intervals and hypothesis tests for μ₁ − μ₂.
考虑两个独立样本：一个来自 N(μ₁, σ₁²) 大小为 n₁，另一个来自 N(μ₂, σ₂²) 大小为 n₂。差值 X̄₁ − X̄₂ 服从正态分布，均值为 μ₁ − μ₂，方差为 (σ₁²/n₁) + (σ₂²/n₂)。即使总体非正态，根据中心极限定理，当样本量大时差值的分布近似正态，从而能够对 μ₁ − μ₂ 构造置信区间和进行假设检验。
9. Practical Examples and Applications | 实际例子与应用
² + w₂²σ₂². In manufacturing, the total length of assembled parts X + Y has variance equal to the sum of individual variances if the parts are produced independently. Such additive noise concepts appear in error propagation.
在金融中，权重为 w₁、w₂ 的两资产组合回报为 R = w₁R₁ + w₂R₂。E(R) = w₁μ₁ + w₂μ₂，若回报独立，Var(R) = w₁²σ₁² + w₂²σ₂²。在制造中，装配部件 X + Y 的总长度，若部件独立生产，其方差等于个体方差之和。这种可加噪声概念出现在误差传递中。
10. Common Mistakes and Pitfalls | 常见错误与陷阱
coefficients: Var(3X) = 9Var(X), not 3Var(X). When combining normals, ensure independence—mixing non-independent normals can yield non-normal results.
一个常见错误是对相关变量使用独立方差公式，遗漏了协方差项。另一个错误是加减标准差而非方差；相加的是方差，而非标准差。学生还常忘记对系数取平方：Var(3X) = 9Var(X)，而非 3Var(X)。组合正态变量时须确保独立——对非独立正态变量混合可能产生非正态结果。
11. Summary and Key Formulas | 总结与关键公式
The core relationships for combining random variables can be summarised concisely. The table below gathers the essential formulas for quick reference.
组合随机变量的核心关系可简洁总结。下表汇总了关键公式以便快速查阅。
| Property | Formula (a, b constants) |
|---|---|
| Expectation | E(aX + bY) = aE(X) + bE(Y) |
| Variance (independent) | Var(aX + bY) = a²Var(X) + b²Var(Y) |
| Variance (general) | Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab Cov(X,Y) |
| Normal combination (independent) | aX + bY ~ N(aμ₁ + bμ₂, a²σ₁² + b²σ₂²) |
| Sample mean variance | Var(X̄) = σ²/n |
12. Exam Tips for IB | IB考试技巧
When tackling IB problems on combined variables, first identify whether independence is given or reasonable. Write down the formulas, taking care to square coefficients. If the distribution is requested, check if variables are normal or if the Central Limit Theorem applies. For hypothesis testing on differences of means, ensure you use the correct variance formula. Practice recognising additive properties for binomial and Poisson distributions, and always double-check units and interpretations.
在解决IB中有关组合变量的问题时，首先确认独立性是否给定或合理。写出公式，注意对系数取平方。若要求分布，检查变量是否正态或中心极限定理是否适用。对均值差进行假设检验时，确保使用正确的方差公式。练习识别二项分布和泊松分布的可加性质，并始终复核单位和解释。
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