Linear Combinations of Normal Variables | 正态变量的线性组合

📚 Linear Combinations of Normal Variables | 正态变量的线性组合

When independent normal variables are added, subtracted, or scaled, the result is again normally distributed. This powerful property enables us to model sums, differences, and averages in statistics and probability, and it forms a cornerstone of inference.

当独立的正态变量进行加减或缩放时，结果仍然服从正态分布。这一强大性质使我们能够对统计学和概率中的总和、差值及平均值进行建模，并且它是统计推断的重要基石。

---

1. Recap of the Normal Distribution | 正态分布回顾

A continuous random variable X is said to be normally distributed with mean μ and variance σ² if its probability density function is bell-shaped and symmetric about μ. We write X ~ N(μ, σ²).

如果连续随机变量 X 的概率密度函数呈钟形且关于 μ 对称，则称 X 服从均值为 μ、方差为 σ² 的正态分布，记作 X ~ N(μ, σ²)。

The parameters μ and σ² fully determine the distribution. The standard deviation is σ. Changing μ shifts the curve left or right; changing σ stretches or squeezes it.

参数 μ 和 σ² 完全决定了分布。标准差为 σ。改变 μ 会使曲线左右平移；改变 σ 会使其拉伸或压缩。

---

2. Independent Random Variables | 独立随机变量

uch variables, the joint probability factorises and the covariance is zero: Cov(X, Y) = 0.

如果两个随机变量 X 和 Y 的任意取值互不影响对方的概率分布，则称它们独立。对于独立变量，联合概率可分解，且协方差为零：Cov(X, Y) = 0。

Throughout this topic we will assume independence unless stated otherwise, because it simplifies the variance rule dramatically.

在本主题中，除非特别说明，我们均假定变量相互独立，因为这将极大简化方差的计算规则。

---

3. Linear Combination Defined | 线性组合的定义

A linear combination of random variables X and Y takes the form aX + bY, where a and b are constants. More generally, for n variables we can write a₁X₁ + a₂X₂ + ... + a_nX_n.

随机变量 X 和 Y 的线性组合具有 aX + bY 的形式，其中 a 与 b 为常数。更一般地，对于 n 个变量可表示为 a₁X₁ + a₂X₂ + ... + a_nX_n。

clude the sum (a = 1, b = 1), the difference (a = 1, b = -1), and weighted averages (a = w₁, b = w₂ where weights sum to 1).

常见的例子包括求和（a = 1, b = 1）、求差（a = 1, b = -1）以及加权平均（a = w₁, b = w₂ 且权重之和为 1）。

---

4. Expectation of a Linear Combination | 线性组合的期望

The expected value of a linear combination follows a simple linearity rule:

E(aX + bY) = a E(X) + b E(Y)

线性组合的期望遵循简单的线性规则：

E(aX + bY) = a E(X) + b E(Y)

This holds for any random variables, whether independent or not, because expectation is a linear operator.

这对任何随机变量都成立，无论它们是否独立，因为期望是线性算子。

When X ~ N(μ_X, σ_X²) and Y ~ N(μ_Y, σ_Y²), we immediately obtain E(aX + bY) = aμ_X + bμ_Y.

当 X ~ N(μ_X, σ_X²) 且 Y ~ N(μ_Y, σ_Y²) 时，直接可得 E(aX + bY) = aμ_X + bμ_Y。

---

5. Variance of a Linear Combination | 线性组合的方差

For independent random variables X and Y, the variance behaves almost as simply as the expectation, but with squared coefficients:

Var(aX + bY) = a² Var(X) + b² Var(Y)

对于独立的随机变量 X 和 Y，方差的计算几乎与期望一样简单，但系数需平方：

Var(aX + bY) = a² Var(X) + b² Var(Y)

If X and Y are not independent, a covariance term must be added: Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab Cov(X, Y). Our focus remains on independent variables where this term vanishes.

如果 X 和 Y 不独立，则需要加上协方差项：Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab Cov(X, Y)。我们着重讨论独立情形，此时该项为零。

ndent X and Y, Var(X - Y) = Var(X) + Var(Y), because the coefficient b = -1 squares to +1.

需要特别注意的是，差的方差并不是方差之差。对于独立的 X 和 Y，Var(X - Y) = Var(X) + Var(Y)，因为系数 b = -1 平方后变为 +1。

---

6. Distribution of the Linear Combination | 线性组合的分布

The central result: any linear combination of independent normal variables is itself normally distributed. That is, if X_i ~ N(μ_i, σ_i²) are independent, then

a₁X₁ + a₂X₂ + ... + a_nX_n ~ N(∑ a_iμ_i, ∑ a_i²σ_i²)

核心结论：独立正态变量的任意线性组合仍然服从正态分布。也就是说，若 X_i ~ N(μ_i, σ_i²) 相互独立，则

a₁X₁ + a₂X₂ + ... + a_nX_n ~ N(∑ a_iμ_i, ∑ a_i²σ_i²)

This property is sometimes called the reproductive property of the normal distribution and is one reason why the normal model is so widely used in statistics.

该性质有时被称为正态分布的再生性，也是正态模型在统计学中广泛使用的原因之一。

---

广告 / Sponsored

7. Example: Sum of Two Normal Variables | 例题：两个正态变量之和

Suppose the weights of apples and oranges in a crate are independent with X ~ N(150, 10²) g and Y ~ N(200, 15²) g. Find the distribution of the total weight T = X + Y.

假设箱子中苹果和橙子的重量相互独立，且 X ~ N(150, 10²) g，Y ~ N(200, 15²) g。求总重量 T = X + Y 的分布。

The mean: E(T) = 150 + 200 = 350 g. The variance: Var(T) = 10² + 15² = 100 + 225 = 325 g². Hence T ~ N(350, 325).

均值：E(T) = 150 + 200 = 350 g。方差：Var(T) = 10² + 15² = 100 + 225 = 325 g²。因此 T ~ N(350, 325)。

The standard deviation is √325 ≈ 18.03 g. Because the sum is normal, we can immediately compute probabilities such as P(T > 370).

标准差为 √325 ≈ 18.03 g。由于总和服从正态，我们可以立即计算 P(T > 370) 等概率。

---

8. Example: Difference of Two Normal Variables | 例题：两个正态变量之差

the same fruits, let D = X - Y be the difference in weight between an apple and an orange.

使用同样的水果，设 D = X - Y 表示苹果与橙子的重量差。

E(D) = 150 - 200 = -50 g, so on average the apple is lighter. The variance is Var(D) = 1²Var(X) + (-1)²Var(Y) = 100 + 225 = 325 g², exactly the same as for the sum.

E(D) = 150 - 200 = -50 g，因此平均而言苹果更轻。方差为 Var(D) = 1²Var(X) + (-1)²Var(Y) = 100 + 225 = 325 g²，与总和的方差完全相同。

Thus D ~ N(-50, 325). The fact that the variance is the sum of the individual variances, not their difference, is a common source of error.

因此 D ~ N(-50, 325)。方差为两者方差之和而非它们的差，这一点是常见错误来源。

---

9. Extension to n Variables | 推广至 n 个变量

e than two independent normal variables, the same rules apply. For instance, the sum S = X₁ + X₂ + ... + X_n has

S ~ N(μ₁ + μ₂ + ... + μ_n, σ₁² + σ₂² + ... + σ_n²)

当存在两个以上的独立正态变量时，同样的规则依然适用。例如，和 S = X₁ + X₂ + ... + X_n 有

S ~ N(μ₁ + μ₂ + ... + μ_n, σ₁² + σ₂² + ... + σ_n²)

If the variables are identically distributed with mean μ and variance σ², the sample mean &bar;X = (X₁ + ... + X_n)/n is also normal: &bar;X ~ N(μ, σ²/n).

若变量同分布且均值为 μ、方差为 σ²，则样本均值 &bar;X = (X₁ + ... + X_n)/n 也服从正态：&bar;X ~ N(μ, σ²/n)。

---

10. Calculating Probabilities Using Standardisation | 通过标准化计算概率

mal distribution of a linear combination L, we can standardise it to the standard normal Z ~ N(0, 1) using Z = (L - μ_L) / σ_L.

一旦得知线性组合 L 的正态分布，我们可以通过 Z = (L - μ_L) / σ_L 将其标准化为标准正态 Z ~ N(0, 1)。

For the total weight T ~ sing: Z = (370 - 350)/√325 ≈ 1.109. Using tables, P(Z > 1.109) ≈ 0.1335.

对于总重量 T ~ N(350, 325)，求 P(T > 370) 需标准化：Z = (370 - 350)/√325 ≈ 1.109。查表得 P(Z > 1.109) ≈ 0.1335。

In examination settings, you may need to work backwards: given a probability, find the unknown mean or variance using inverse normal calculations.

在考试中，你可能需要反向求解：给定概率，通过逆正态计算求出未知的均值或方差。

---

11. Common Mistakes | 常见错误

r(X) - Var(Y). The correct rule for independent variables is Var(X - Y) = Var(X) + Var(Y).

错误一：用减号计算方差。许多学生错误地写出 Var(X - Y) = Var(X) - Var(Y)。独立变量的正确规则是 Var(X - Y) = Var(X) + Var(Y)。

Mistake 2: Forgetting that the sum of normal variables is normal only if they are independent (or jointly normal). If dependence is present, the combination may not be normal.

错误二：忽略了正态变量之和仅在变量独立（或联合正态）时才服从正态分布。若存在相依性，组合可能不再是正态。

Mistake 3: Mixing up standard deviation and variance. Always square the coefficients for variance, but standard deviation is the square root of that sum.

错误三：混淆标准差与方差。系数在方差中需平方，而标准差是方差和的平方根。

---

12. Summary and Key Points | 总结与要点

Linear combinations of independent normal variables are normal. The mean is the same linear combination of the individual means, and the variance is the sum of the squares of the coefficients times the individual variances.

独立正态变量的线性组合仍为正态。均值为各变量均值的相同线性组合，方差为系数平方与各自方差乘积的总和。

This result underpins the distribution of sums, differences, sample means, and many estimators. Always check independence before applying the variance formula and remember that variance always adds when subtracting.

这一结果支撑了总和、差值、样本均值以及许多估计量的分布。务必在应用方差公式前确认独立性，并牢记做减法时方差仍需相加。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）