IB Mathematics: Combinations of Random Variables | IB数学：随机变量的组合方式

📚 IB Mathematics: Combinations of Random Variables | IB数学：随机变量的组合方式

In IB Mathematics, random variables rarely appear alone. Exam questions often ask you to combine two or more random variables, or to study a function of a single random variable. Understanding how expectations, variances, and distributions behave under these combinations is essential for statistics, probability, and data-analysis questions.

在IB数学中，随机变量很少单独出现。考试题目常常要求你组合两个或多个随机变量，或者研究单个随机变量的函数。理解期望、方差和分布在组合运算下如何变化，是解决统计、概率和数据分析问题的关键。

1. The Idea of Combining Random Variables | 组合随机变量的基本思想

A combination of random variables is a new random variable formed by adding, subtracting, or scaling existing variables. For example, if X is the weight of a box and Y is the weight of its contents, then X + Y is the total weight, and X − Y could represent the weight of the box alone if Y is measured independently.

随机变量的组合，就是通过加法、减法或缩放已有的变量，从而构成一个新的随机变量。例如，如果X是箱子的重量，Y是箱内物品的重量，那么X + Y就是总重量；如果Y是独立测量得到的，那么X − Y可以表示空箱重量。

These combinations are not just arithmetic exercises. They model real situations such as portfolio returns, total scores from multiple test papers, or the difference between two measurements. IB examiners expect you to know both the mean and variance rules for such combinations.

这种组合不仅仅是算术练习。它们可以刻画真实情境，例如投资组合收益、多张试卷的总分，或者两次测量之间的差值。IB考官希望你能够掌握这类组合的均值规则和方差规则。

2. Expectation of Linear Combinations | 线性组合的期望

The expectation operator is linear. For any random variables X and Y, and any constants a, b, and c:

期望运算具有线性性质。对于任意随机变量X和Y，以及任意常数a、b、c，都有：

E(aX + bY + c) = aE(X) + bE(Y) + c

This rule holds regardless of whether X and Y are independent. It works because expectation is an average, and averages respect addition and constant multiples.

这个公式无论X和Y是否独立都成立。原因在于期望本质上是一种平均，而平均运算满足加法与常数倍数的分配性质。

For example, if E(X) = 3 and E(Y) = 5, then E(2X − Y + 1) = 2(3) − 5 + 1 = 2. The constants can be positive or negative, but the logic stays the same.

例如，若E(X) = 3，E(Y) = 5，则E(2X − Y + 1) = 2(3) − 5 + 1 = 2。常数可以是正数或负数，但计算逻辑不变。

3. Variance of Linear Combinations | 线性组合的方差

Variance behaves differently from expectation. When you add independent variables, their variances add, but when you multiply a variable by a constant, the constant is squared:

方差的行为与期望不同。当独立变量相加时，方差相加；但当变量乘以常数时，常数要平方：

Var(aX + bY) = a²Var(X) + b²Var(Y) if X and Y are independent

This means that adding a constant to a variable does not change its variance: Var(X + c) = Var(X). Scaling a variable by a factor a multiplies the standard deviation by |a|, but multiplies the variance by a².

这意味着给变量加上一个常数不会改变方差：Var(X + c) = Var(X)。将变量乘以因子a时，标准差乘以|a|，而方差则乘以a²。

A common IB trap is to write Var(2X) = 2Var(X). The correct result is Var(2X) = 4Var(X). If the standard deviation of X is 3, then the standard deviation of 2X is 6, not 3√2.

一个常见的IB陷阱是写成Var(2X) = 2Var(X)。正确结果是Var(2X) = 4Var(X)。如果X的标准差为3，那么2X的标准差是6，而不是3√2。

4. Independence and Covariance | 独立性与协方差

If X and Y are not independent, an extra covariance term appears:

如果X和Y不独立，就会多出一个协方差项：

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

When X and Y are independent, Cov(X, Y) = 0, so the formula reduces to the simple addition of variances. When they are positively correlated, the variance of the sum is larger; when negatively correlated, it is smaller.

当X和Y独立时，Cov(X, Y) = 0，因此公式退化为方差直接相加。当它们正相关时，和的方差更大；当它们负相关时，和的方差更小。

For the difference of two variables, the covariance term is subtracted:

对于两个变量之差，协方差项前面是减号：

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

Note that even for independent variables, Var(X − Y) = Var(X) + Var(Y), not Var(X) − Var(Y). Subtraction of independent variables still adds their variances.

注意，即使变量独立，Var(X − Y) = Var(X) + Var(Y)，而不是Var(X) − Var(Y)。独立变量相减时，方差仍然相加。

5. Sums of Independent Normal Variables | 独立正态变量之和

If X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²) are independent, then their sum is also normal:

若X ~ N(μ₁, σ₁²)，Y ~ N(μ₂, σ₂²)，且二者独立，那么它们的和仍然是正态分布：

X + Y ~ N(μ₁ + μ₂, σ₁² + σ₂²)

Similarly, their difference is normal:

同理，它们的差也服从正态分布：

X − Y ~ N(μ₁ − μ₂, σ₁² + σ₂²)

Notice that the mean is subtracted for a difference, but the variance is never subtracted. Both sum and difference use a plus sign for the variance.

注意：求差时均值做减法，但方差从不做减法。无论是和还是差，方差都用加号。

This result extends to any linear combination of independent normal variables. For example, 2X + 3Y is normal with mean 2μ₁ + 3μ₂ and variance 4σ₁² + 9σ₂².

这一结论可以推广到独立正态变量的任意线性组合。例如，2X + 3Y服从正态分布，其均值为2μ₁ + 3μ₂，方差为4σ₁² + 9σ₂²。

6. Multiplying by a Constant | 随机变量乘以常数

Multiplying a random variable by a constant changes both the centre and the spread of its distribution. If X ~ N(μ, σ²), then:

将随机变量乘以常数，会同时改变分布的中心和离散程度。若X ~ N(μ, σ²)，则：

aX + b ~ N(aμ + b, a²σ²)

For example, if the temperature in Celsius is C ~ N(20, 4), then the temperature in Fahrenheit is F = (9/5)C + 32. Hence F is normal with mean (9/5)(20) + 32 = 68 and variance (9/5)² × 4 = 12.96.

例如，若以摄氏度表示的温度为C ~ N(20, 4)，那么以华氏度表示的温度为F = (9/5)C + 32。因此F服从正态分布，均值为(9/5)(20) + 32 = 68，方差为(9/5)² × 4 = 12.96。

This rule is crucial when converting measurement units. Always square the multiplier when computing variance, not when computing the mean.

这个规则在单位换算时非常重要。计算方差时一定要把倍数平方，而计算均值时不需要平方。

7. The Sample Mean | 样本均值

The sample mean is a special linear combination of independent, identically distributed variables. If X₁, X₂, …, Xₙ are independent variables each with mean μ and variance σ², then the sample mean is:

样本均值是独立同分布变量的一种特殊线性组合。若X₁, X₂, …, Xₙ是独立变量，且每个变量的均值为μ、方差为σ²，那么样本均值为：

X̄ = (X₁ + X₂ + … + Xₙ) / n

Its expectation is:

其期望为：

E(X̄) = μ

Its variance is:

其方差为：

Var(X̄) = σ² / n

Therefore the standard deviation of the sample mean is σ/√n, often called the standard error of the mean. As n increases, the sample mean becomes less variable and more concentrated around μ.

因此样本均值的标准差为σ/√n，通常称为均值的标准误。随着n增大，样本均值的波动变小，也越来越集中到μ附近。

8. Standardised Combinations and Z-Scores | 标准化组合与Z分数

A useful technique in IB Statistics is to standardise a combination so that it follows a standard normal distribution N(0, 1). For a single random variable X:

IB统计中的一个常用技巧是将组合标准化，使其服从标准正态分布N(0, 1)。对于单个随机变量X：

Z = (X − μ) / σ

For the sample mean X̄, the standardised variable is:

对于样本均值X̄，标准化后的变量为：

Z = (X̄ − μ) / (σ / √n)

These z-scores allow you to use the standard normal table or a GDC to find probabilities. The key is to identify which standard deviation applies: σ for the original variable, and σ/√n for the sample mean.

这些z分数使你可以使用标准正态分布表或图形计算器来求概率。关键是要分清该用哪个标准差：原始变量用σ，样本均值用σ/√n。

If the original variables are not normal but n is large, the Central Limit Theorem states that X̄ is approximately normal. Many IB questions explicitly ask you to state this approximation before doing the probability calculation.

如果原始变量不服从正态分布，但n很大，中心极限定理指出X̄近似服从正态分布。许多IB题目会明确要求你说明这一近似，然后再进行概率计算。

9. Applications to Confidence Intervals | 在置信区间中的应用

Combinations of random variables are the foundation of confidence intervals. For a population mean μ with known σ, a 95% confidence interval based on a sample of size n is:

随机变量的组合是置信区间的基础。对于均值μ，若σ已知，基于容量为n的样本的95%置信区间为：

X̄ ± 1.96 × σ / √n

The margin of error is 1.96 × σ/√n. This margin is half the width of the interval and comes directly from the standard deviation of the sample mean.

误差幅度为1.96 × σ/√n。这个幅度是区间宽度的一半，直接来自样本均值的标准差。

If σ is unknown, IB students often replace σ with the sample standard deviation s and use a t-distribution. However, for large samples, the normal approximation is acceptable.

如果σ未知，IB学生通常用样本标准差s代替σ，并使用t分布。不过对于大样本，正态近似也可以接受。

Combination	Mean	Variance (independent variables)
X + Y	μ₁ + μ₂	σ₁² + σ₂²
X − Y	μ₁ − μ₂	σ₁² + σ₂²
aX + b	aμ + b	a²σ²
X̄ = (ΣXᵢ)/n	μ	σ² / n

10. Common Pitfalls | 常见易错点

Pitfall 1: Adding standard deviations directly. If X and Y are independent, SD(X + Y) = √(Var(X) + Var(Y)), not SD(X) + SD(Y). Standard deviations do not add.

易错点1：直接相加标准差。若X和Y独立，SD(X + Y) = √(Var(X) + Var(Y))，而不是SD(X) + SD(Y)。标准差不能直接相加。

Pitfall 2: Forgetting that variance is squared. Many students write Var(3X) = 3Var(X). The correct form is Var(3X) = 9Var(X).

易错点2：忘记方差需要平方。很多学生会写成Var(3X) = 3Var(X)。正确形式是Var(3X) = 9Var(X)。

Pitfall 3: Subtracting variances for X − Y. Even for a difference, the variances add: Var(X − Y) = Var(X) + Var(Y) when X and Y are independent.

易错点3：对X − Y做方差减法。即使是求差，方差也相加：当X和Y独立时，Var(X − Y) = Var(X) + Var(Y)。

Pitfall 4: Assuming independence without justification. If the question says two variables are dependent, you must include the covariance term or use the provided correlation coefficient.

易错点4：没有依据就假设独立。如果题目说明两个变量相关，你必须加入协方差项，或使用题目给出的相关系数。

Pitfall 5: Confusing X with X̄. The variance of a single observation is σ², but the variance of the sample mean is σ²/n. Trying to standardise the sample mean using σ instead of σ/√n leads to wrong probabilities.

易错点5：混淆X与X̄。单个观测值的方差是σ²，但样本均值的方差是σ²/n。若在标准化样本均值时误用σ而不是σ/√n，会得到错误的概率。

11. Summary | 总结

Combinations of random variables are governed by two main rules: expectations always add with coefficients, and variances add only when variables are independent, with coefficients squared. The normal distribution remains normal under any linear combination, and the sample mean has a smaller variance of σ²/n.

随机变量的组合由两条主要规则控制：期望总是按系数相加；方差只有在变量独立时才直接相加，且系数需要平方。正态分布在任意线性组合下仍是正态分布，而样本均值的方差为σ²/n，比单个观测值的方差更小。

Master these rules, practise standardising variables, and use the table of formulas carefully. With consistent revision, combination-of-variables questions become some of the most predictable marks in the IB mathematics exams.

掌握这些规则，勤加练习标准化变量，并谨慎使用公式表，组合随机变量的题目就会成为IB数学考试中最好拿分、最稳定的题型之一。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IB Mathematics: Combinations of Random Variables | IB数学：随机变量的组合方式