Combinations of Random Variables | 随机变量的组合

📚 Combinations of Random Variables | 随机变量的组合

In Edexcel A-Level Mathematics, the topic ‘combinations of random variables’ asks what happens when you add, subtract, or scale random variables. The rules are simple but very easy to misuse: expectations add linearly, while variances have squared coefficients and always add for independent variables, even when you subtract.

在 Edexcel A-Level 数学中，“随机变量的组合”这一主题考察的是当你对随机变量进行加减或缩放时会发生什么。规则简单但很容易用错：期望可以线性相加；而对于独立的随机变量，方差带有系数平方，并且即使做减法，方差仍然相加。

1. The Core Idea of Combining Random Variables | 随机变量组合的核心思想

Suppose X and Y are random variables, such as two independent measurements or scores. A combination is any expression like aX + bY + c. We cannot simply transform individual probabilities directly; instead we use rules for expectation and variance. Edexcel questions usually state independence when it is needed.

假设 X 和 Y 是随机变量，例如两个独立的测量值或得分。组合就是像 aX + bY + c 这样的表达式。我们不能直接单独变换每个变量的概率，而是要使用期望和方差的运算法则。Edexcel 题目通常会在需要时明确说明变量是否独立。

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

Expectation is linear. For constants a, b and c, the expectation of aX + bY + c is aE(X) + bE(Y) + c. This holds whether or not X and Y are independent. The constant c shifts the mean but has no effect on spread. For a single variable, E(aX + c) = aE(X) + c.

期望具有线性性质。对于常数 a、b 和 c，aX + bY + c 的期望为 aE(X) + bE(Y) + c。无论 X 与 Y 是否独立，这个公式都成立。常数 c 只会平移均值，对离散程度没有影响。对于单个变量，E(aX + c) = aE(X) + c。

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

For example, if E(X) = 8 and E(Y) = 5, then E(3X − 2Y + 4) = 3(8) − 2(5) + 4 = 18.

例如，若 E(X) = 8，E(Y) = 5，则 E(3X − 2Y + 4) = 3(8) − 2(5) + 4 = 18。

3. Variance of a Linear Combination for Independent Variables | 独立变量线性组合的方差

If X and Y are independent, then Var(aX ± bY) = a²Var(X) + b²Var(Y). Notice that the coefficients are squared, and the variance of a constant is zero. If the variables are not independent, this simple rule is incomplete because covariance terms appear. Edexcel will specify independence when this rule is required.

如果 X 和 Y 独立，则 Var(aX ± bY) = a²Var(X) + b²Var(Y)。注意系数需要平方，常数的方差为零。如果变量不独立，这个简单公式就不完整，因为会出现协方差项。Edexcel 会在需要使用该规则时明确说明变量独立。

Var(aX ± bY) = a²Var(X) + b²Var(Y)

To find the standard deviation, calculate the variance first and then take the square root at the end.

要求标准差时，应先计算方差，最后再开平方。

4. Why the Variance Adds for X − Y | 为什么 X − Y 的方差仍然是相加

A common mistake is to think that X − Y has variance Var(X) − Var(Y). In fact Var(X − Y) = Var(X) + Var(Y) for independent X and Y, because Var(−Y) = (−1)²Var(Y) = Var(Y). The minus sign is squared away. Taking the difference between two uncertain quantities gives a result that is more spread out than either original variable.

一个常见错误是认为 X − Y 的方差为 Var(X) − Var(Y)。事实上，对于独立的 X 和 Y，Var(X − Y) = Var(X) + Var(Y)，因为 Var(−Y) = (−1)²Var(Y) = Var(Y)。负号在平方后消失。对两个不确定量进行相减，得到的结果会比原来任何一个变量都更分散。

5. Linear Transformations and Repeated Observations | 线性变换与重复观测

If Y = aX + b, then E(Y) = aE(X) + b and Var(Y) = a²Var(X). For n independent observations X₁, X₂, …, Xₙ with the same distribution as X, the total T = X₁ + X₂ + … + Xₙ has E(T) = nE(X) and Var(T) = nVar(X). The sample mean X̄ = T/n has E(X̄) = E(X) and Var(X̄) = Var(X)/n.

如果 Y = aX + b，则 E(Y) = aE(X) + b，Var(Y) = a²Var(X)。对于 n 个独立观测值 X₁, X₂, …, Xₙ，它们与 X 具有相同分布，则总和 T = X₁ + X₂ + … + Xₙ 满足 E(T) = nE(X)，Var(T) = nVar(X)。样本均值 X̄ = T/n 满足 E(X̄) = E(X)，Var(X̄) = Var(X)/n。

T = X₁ + X₂ + … + Xₙ
E(T) = nE(X), Var(T) = nVar(X)
X̄ = T/n, E(X̄) = E(X), Var(X̄) = Var(X)/n