📚 Linear Combinations of Random Variables | 随机变量的线性组合
In A-Level Further Mathematics (Edexcel), the topic of linear combinations of random variables is a cornerstone of Further Statistics 1. Understanding how to derive the mean and variance of sums and multiples of random variables is essential for tackling real-world problems, from portfolio risk to quality control. This article explores the key rules, proofs, and applications, including the special cases of normal and Poisson distributions.
在A-Level进阶数学（Edexcel）中，随机变量的线性组合是Further Statistics 1的核心主题。掌握如何推导多个随机变量之和与数乘的期望和方差，对于解决从投资组合风险到质量控制的现实问题至关重要。本文将探讨关键规则、证明及其应用，包括正态分布和泊松分布的特殊情形。
1. Introduction to Linear Combinations | 线性组合简介
A linear combination of random variables X and Y takes the form aX + bY, where a and b are constants. If we have more than two variables, we can extend to Σ aᵢ Xᵢ. The primary goal is to find expressions for the expectation and variance of such combinations given the properties of the individual variables.
随机变量 X 和 Y 的线性组合具有形式 aX + bY，其中 a 和 b 为常数。如果不止两个变量，可以推广到 Σ aᵢ Xᵢ。主要目标是在给定单个变量性质的前提下，求出这类组合的期望和方差的表达式。
Common scenarios include the total score of multiple dice rolls, the total resistance of resistors in series (with tolerance), and the combined profit from different business units. In each case, we model the total as a linear combination.
常见情形包括多次掷骰子的总得分、串联电阻的总电阻（含公差），以及不同业务部门的总利润。在每种情况下，我们都将总数建模为线性组合。
2. Expectation of a Linear Combination | 线性组合的期望
For any random variables X and Y, and constants a and b, the expectation of a linear combination is given by:
E(aX + bY) = aE(X) + bE(Y)
This result holds even if X and Y are not independent. It follows directly from the linearity of the expectation operator.
对于任意随机变量 X 和 Y 以及常数 a 和 b，线性组合的期望为：E(aX + bY) = aE(X) + bE(Y)。该结论即便在 X 和 Y 不独立时也成立，直接源于期望算子的线性性质。
. This property is widely used in statistics, for instance, when calculating the expected total of a sample sum.
更一般地，E(Σ aᵢ Xᵢ) = Σ aᵢ E(Xᵢ)。这一性质在统计学中广泛应用，例如计算样本总和的期望时。
3. Variance of a Linear Combination for Independent Variables | 独立变量线性组合的方差
When X and Y are independent, the variance simplifies to:
Var(aX + bY) = a² Var(X) + b² Var(Y)
ppears. This extends to any number of independent variables: Var(Σ aᵢ Xᵢ) = Σ aᵢ² Var(Xᵢ).
当 X 和 Y 独立时，方差简化为：Var(aX + bY) = a² Var(X) + b² Var(Y)。注意常数 a 和 b 需要平方。独立性使协方差项消失。这可以推广到任意多个独立变量：Var(Σ aᵢ Xᵢ) = Σ aᵢ² Var(Xᵢ)。
2Y) = 3² × 4 + (-2)² × 9 = 36 + 36 = 72. The negative sign on b disappears after squaring.
例如，若 X 和 Y 独立且 Var(X)=4, Var(Y)=9，则 Var(3X - 2Y) = 3² × 4 + (−2)² × 9 = 36 + 36 = 72。b 的负号平方后消失。
4. General Variance Formula with Covariance | 含协方差的一般方差公式
If X and Y are not independent, we must include covariance:
Var(aX + bY) = a² Var(X) + b² Var(Y) + 2ab Cov(X, Y)
ndent variables, Cov(X,Y)=0, and the formula reduces to the one for independent variables.
如果 X 和 Y 不独立，则必须包含协方差：Var(aX + bY) = a² Var(X) + b² Var(Y) + 2ab Cov(X,Y)。协方差衡量两个变量共同变化的程度，可正可负。对于独立变量，Cov(X,Y)=0，公式即退化为独立变量的情形。
estions, you may be given the covariance directly or have to derive it from joint probabilities.
协方差可通过 Cov(X,Y) = E(XY) − E(X)E(Y) 计算
Published by TutorHao | A-Level 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