📚 Combinations of Random Variables | 随机变量的组合
In A-Level Statistics, we rarely look at a single random variable in isolation. Real-world problems often require us to combine variables — for instance, the total mass of several parcels, the difference between two machine outputs, or the average of repeated measurements. This chapter develops the rules for expectations, variances, and distributions of linear combinations such as aX + bY + c. Mastering these rules is essential for tackling topics like the distribution of sample means and for solving exam questions on sums and differences of independent normal, Poisson, or binomial variables.
在A-Level统计学中，我们很少孤立地看待一个随机变量。实际问题通常需要我们组合变量——例如多个包裹的总质量、两台机器产出之差、或重复测量的平均值。本章将推导线性组合（如 aX + bY + c）的期望、方差和分布的规则。掌握这些规则对于处理样本均值的分布以及解答关于独立正态、泊松或二项变量的和与差的考题至关重要。
1. Introduction to Combinations of Random Variables | 随机变量组合简介
near combination: W = aX + bY + c, where X and Y are random variables, and a, b, c are real numbers. We can also form sums (X + Y), differences (X - Y), scaled versions (3X), and samples mean (X̄ = (X₁ + X₂ + … + Xₙ)/n). The key questions are: what is the expected value of W? What is its variance? And, if X and Y follow known distributions, what distribution does W follow?
随机变量的组合是指包含两个或多个随机变量与常数构成的任何函数。最常见的形式为线性组合：W = aX + bY + c，其中X和Y是随机变量，a、b、c是实数。我们还可以构造和（X + Y）、差（X - Y）、数乘（3X）以及样本均值（X̄ = (X₁ + X₂ + … + Xₙ)/n）。关键问题是：W的期望是什么？它的方差是多少？如果X和Y服从已知分布，W服从什么分布？
2. Expected Value of a Linear Combination | 线性组合的期望
The expectation operator E is linear. For any random variables X and Y, and constants a, b, c, we have E(aX + b) = aE(X) + b. More generally, E(aX + bY + c) = aE(X) + bE(Y) + c. This holds regardless of whether X and Y are independent. For the sample mean X̄ = (X₁ + X₂ + … + Xₙ)/n, this linearity gives E(X̄) = μ, where each Xᵢ has the same mean μ.
期望算子E是线性的。对于任意随机变量X和Y，以及常数a、b、c，有E(aX + b) = aE(X) + b。更一般地，E(aX + bY + c) = aE(X) + bE(Y) + c。这个结论不要求X和Y独立。对于样本均值X̄ = (X₁ + X₂ + … + Xₙ)/n，利用线性性可得E(X̄) = μ，其中每个Xᵢ具有相同的均值μ。
3. Variance of a Linear Combination | 线性组合的方差
The variance of a linear function of a single variable is Var(aX + b) = a² Var(X). The constant b disappears because adding a constant does not change spread. For two variables, the general formula includes covariance: Var(aX + bY + c) = a² Var(X) + b² Var(Y) + 2ab Cov(X, Y). If X and Y are independent, then Cov(X, Y) = 0, so the variance simplifies to Var(aX + bY) = a² Var(X) + b² Var(Y). Notice that for the difference of independent variables, Var(X - Y) = Var(X) + Var(Y) – the variances always add, even when subtracting.
单个变量线性函数的方差为Var(aX + b) = a² Var(X)。常数b不会出现，因为加上常数不改变离散程度。对于两个变量，一般公式包含协方差：Var(aX + bY + c) = a² Var(X) + b² Var(Y) + 2ab Cov(X, Y)。若X和Y独立，则Cov(X, Y) = 0，于是方差简化为Var(aX + bY) = a² Var(X) + b² Var(Y)。注意，对于独立变量之差，Var(X - Y) = Var(X) + Var(Y)——即使相减，方差也总是相加。
4. Independent Random Variables | 独立随机变量
Two random variables are independent if the occurrence of any value of X does not affect the probability distribution of Y. For independent variables, E(XY) = E(X)E(Y) ently, the variance of a sum or difference of independent variables becomes simply the sum of their variances, scaled by the squares of any multipliers. A useful summary is given in the table below for independent X and Y with constants a, b.
若两个随机变量独立，即X的任何取值不会影响Y的概率分布，则E(XY) = E(X)E(Y)，从而Cov(X, Y) = 0。因此，独立变量和或差的方差简化为它们各自方差乘以系数平方后的和。下表总结了独立X和Y乘以常数a、b的情形。
| Expression (独立表达式) | Variance (方差) |
| aX + b | a² Var(X) |
| X + Y | Var(X) + Var(Y) |
| X - Y | Var(X) + Var(Y) |
| aX + bY | a² Var(X) + b² Var(Y) |
If not, you may need to use covariance or work with joint distributions. The sample mean X̄, based on independent, identically distributed observations X₁, …, Xₙ, has variance Var(X̄) = σ² / n.
在考试题中，你必须首先确认独立性是否给出或隐含。若否，你可能需要使用协方差或联合分布。基于独立同分布观测值X₁, …, Xₙ的样本均值X̄，其方差为Var(X̄) = σ² / n。
5. General Variance Formula with Covariance | 含协方差的一般方差公式
When X and Y are not necessarily independent, we must include their covariance, defined as Cov(X, Y) = E[(X - μx)(Y - μy)] = E(XY) - E(X)E(Y). Covariance measures the degree to which X and Y vary together. It can be positive (both tend to increase together), negative (one increases as the other decreases), or zero. The full variance formula for a linear combination is Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab Cov(X, Y) variables are independent, Cov(X, Y) = 0 and the formula reduces to the independent case.
当X和Y不一定独立时，我们必须包含它们的协方差，其定义为Cov(X, Y) = E[(X - μx)(Y - μy)] = E(XY) - E(X)E(Y)。协方差衡量X和Y一同变异的程度。它可以是正（两者同时增大）、负（一个增大另一个减小）或零。线性组合的完整方差公式为Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab Cov(X, Y)。如果已知Cov(X, Y)，你可以计算精确方差；若变量独立，Cov(X, Y) = 0，公式退化为独立情形。
6. Linear Combinations of Normal Distributions | 正态分布的线性组合
If X ~ N(μx, σx²) and Y ~ N(μy, σy Y also follows a normal distribution. The mean and variance of W are obtained from the linear rules: W ~ N(aμx + bμy, a²σx² + b²σy²). This property is extremely useful because we can transform N(μ, σ²) to the standard normal Z distribution to calculate probabilities. It also means that the sum or difference of independent normals is normal, and the sample mean from a normal population is exactly normal: X̄ ~ N(μ, σ²/n).
若X ~ N(μx, σx²
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