📚 Investigation 1 - E(aX+b) and Var(aX+b) | 探究 1：线性变换的期望与方差
his investigation explores how such a linear transformation affects the expected value (mean) and the variance of the original random variable X. Understanding these rules is fundamental for data standardisation, linear regression, and any situation where we change the units or reference point of measurement.
在概率与统计中，我们经常需要对随机变量进行尺度和位置的变换，从而产生一个新的变量 Y = aX + b。本探究将探讨这种线性变换如何影响原始随机变量 X 的期望值（均值）和方差。理解这些规则对于数据标准化、线性回归以及任何涉及改变测量单位或参考点的情况都至关重要。
1. Setting the Scene: Linear Transformation of a Random Variable | 情境设定：随机变量的线性变换
bled; if a = 0.5, they are halved), while b simply shifts every outcome by a fixed amount.
设 X 是一个具有已知概率分布、期望 E(X) 和方差 Var(X) 的随机变量。线性变换对 X 的每一个取值应用规则 Y = aX + b，其中 a 和 b 是常数。常数 a 充当缩放因子（若 a = 2，数值加倍；若 a = 0.5，数值减半），而 b 只是将所有结果平移固定的量。
This transformation is linear because the graph of Y against X is a straight line with slope a and intercept b. In this investigation, our aim is to find general expressions for E(Y) and Var(Y) in terms of E(X), Var(X), a and b, without needing to re-derive everything from first principles each time.
这种变换是线性的，因为 Y 对 X 的图像是一条斜率为 a、截距为 b 的直线。本次探究的目标是用 E(X)、Var(X)、a 和 b 表达出 E(Y) 和 Var(Y) 的通式，从而避免每次都从基本原理重新推导。
2. Expected Value of a Linear Function | 线性函数的期望
To find E(aX + b), we use the linearity property of expectation. For any random variable X and constants a, b, the expected value satisfies:
为了求 E(aX + b)，我们利用期望的线性性质。对于任意随机变量 X 和常数 a, b，期望满足：
E(aX + b) = aE(X) + b
ince the total probability sums to 1. A similar integration argument works for continuous variables.
该结果对离散和连续随机变量均成立。在离散情况下，E(Y) = Σ (ax + b) P(X = x) = a Σ x P(X = x) + b Σ P(X = x) = aE(X) + b，因为总概率之和为 1。对连续变量也可用类似的积分论证。
The interpretation is straightforward: multiplying every value by a scales the mean by the same factor, and adding b shifts the mean by exactly b. This is exactly what we would intuitively expect from a linear change of units.
解释起来很直接：将每个数值乘以 a，均值就按相同因子缩放；而加上 b 则使均值恰好平移 b。这与我们对单位线性变化的直观预期完全一致。
3. Intuitive Understanding Through Shift and Scale | 通过平移与缩放建立直观理解
s temperature is 20 °C, then the mean Fahrenheit temperature is 1.8 × 20 + 32 = 68 °F. The 'shape' of the distribution remains the same; it is simply relocated and stretched.
设想 X 表示摄氏温度，我们想转换为华氏温度，使用 Y = 1.8X + 32。如果平均摄氏温度为 20 °C，那么平均华氏温度就是 1.8 × 20 + 32 = 68 °F。分布的“形状”保持不变，只是位置移动并拉伸了。
The constant b moves the centre of the distribution left or right without altering its spread. The constant a stretches or compresses the values around the new centre, which will affect the variance – and that is where caution is needed.
常数 b 将分布的中心向左或向右移动，而不改变其离散程度。常数 a 则围绕新中心拉伸或压缩数值，这会影响方差——这正是需要小心的地方。
4. Variance of a Linear Function | 线性函数的方差
iance, the rule is different from the expectation. The variance of Y = aX + b is given by:
对于方差，规则与期望不同。Y = aX + b 的方差由下式给出：
Var(aX + b) = a² Var(X)
Notice that the constant b disappears. Variance measures the average squared deviation from the mean. Adding a constant to all data points shifts the mean but does not change the deviations from the mean. Subtracting the new mean cancels out b entirely.
注意常数 b 消失了。方差衡量的是与均值的平方偏差的平均。对所有数据点加上一个常数，虽然移动了均值，但不改变各点与均值的偏差。减去新均值后，b 完全被抵消。
However, multiplying by a changes the deviations from the mean by a factor of a, and when we square these deviations, we obtain a² times the original squared deviations. Hence the variance is multiplied by a².
然而，乘以 a 会使与均值的偏差变为原来的 a 倍，而这些偏差平方后，就得到原平方偏差的 a² 倍。因此方差被乘以 a²。
5. Proof Using Variance Formula | 使用方差公式的证明
A formal proof uses the definition Var(Y) = E(Y²) – [E(Y)]². Start with Y = aX + b:
正式证明用到定义 Var(Y) = E(Y²) – [E(Y)]²。从 Y = aX + b 开始：
E(Y²) = E[(aX + b)²] = E(a²X² + 2abX + b²) = a²E(X²) + 2abE(X) + b²
[E(Y)]² = [aE(X) + b]² = a²[E(X)]² + 2abE(X) + b²
Subtracting, the 2abE(X) and b² terms cancel, leaving:
相减后，2abE(X) 和 b² 项抵消，剩下：
Var(Y) = a²E(X²) – a²[E(X)]² = a²[E(X²) – (E(X))²] = a²Var(X)
This proof does not assume any particular distribution, only the properties of expectation. It confirms that the variance is unaffected by the location shift b and scaled by the square of a.
该证明不假设任何特定的分布，只用到了期望的性质。它证实了方差不因位置平移 b 而改变，且按 a 的平方缩放。
6. Why Adding b Does Not Affect Spread | 为何加 b 不影响离散程度
ations from the mean are still ±3, ±1, so the variance remains 5. The spread is unchanged.
考虑两组数据：数据集 A: {2, 4, 6, 8}，以及将每个数值加 10 得到的数据集 B: {12, 14, 16, 18}。A 的均值是 5，方差为 5；B 的均值是 15，但与均值的偏差仍为 ±3、±1，因此方差仍是 5。离散程度没有变化。
(X). This property is often called translation invariance of variance. It reminds us that variance is purely a measure of dispersion, not of location.
用概率语言来说，Var(X + b) = Var(X)。这个性质通常被称为方差的平移不变性。它提醒我们，方差纯粹是离散程度的度量，与位置无关。
7. Effect of the Scaling Factor a on Variance | 缩放因子 a 对方差的影响
When we multiply X by a, the spread is stretched by a factor of |a| For example, if a = 3, then Var(3X) = 9 Var(X). If a = 1/2, then Var(X/2) = (1/4) Var(X).
当我们把 X 乘以 a，在原始单位意义上，离散程度被拉伸了 |a| 倍。但由于方差使用平方单位，它会乘上 a²。例如，若 a = 3，则 Var(3X) = 9 Var(X)。若 a = 1/2，则 Var(X/2) = (1/4) Var(X)。
A useful consequence is the effect on standard deviation: σ(aX + b) = |a| This relationship is often used when constructing confidence intervals after scaling data.
一个有用的推论是对标准差的影响：σ(aX + b) = |a| σ(X)。这里需要取 a 的绝对值，因为负的 a 会反转分布，但离散程度不变。这个关系在缩放数据后构建置信区间时经常用到。
8. Worked Examples with Specific Distributions | 具体分布实例演示
Let X follow a normal distribution with mean 50 and variance 16, i.e. X ~ N(50, 16). Suppose we define Y = 2X + 10. Then:
设 X 服从均值为 50、方差为 16 的正态分布，即 X ~ N(50, 16)。假设我们定义 Y = 2X + 10。则：
E(Y) = 2E(X) + 10 = 2 × 50 + 10 = 110
Var(Y) = 2² × 16 = 64, so σ(Y) = √64 = 8
Y is also normally distributed because a linear transformation of a normal variable remains normal. Thus Y ~ N(110, 64). The probability calculations for Y can be standardised using the z-score formula in the usual way.
Y 也是正态分布，因为正态变量的线性变换仍为正态。因此 Y ~ N(110, 64)。对 Y 的概率计算可以像往常一样使用 z 分数标准化。
| Original X | Transformed Y = aX + b | E(Y) | Var(Y) | σ(Y) |
|---|---|---|---|---|
| N(μ, σ²) | aX + b | aμ + b | a²σ² | |a|σ |
| Binomial(n, p) | aX + b | anp + b | a²np(1-p) | |a|√(np(1-p)) |
This table summarises how the mean and variance transform for any distribution. The rules are universal and extremely powerful for simplifying complex problems.
该表总结了均值和方差在任何分布下的变换方式。这些规则是普适的，在简化复杂问题时威力巨大。
9. Application: Standardising Scores and Data | 应用：分数与数据的标准化
e recognise this as a linear transformation with a = 1/σ and b = -μ/σ. Applying our rules:
一个重要的应用是生成 z 分数。给定均值为 μ、标准差为 σ 的 X，标准化变量 Z = (X - μ)/σ 的均值为 0、方差为 1。我们将其视为 a = 1/σ、b = -μ/σ 的线性变换。应用我们的规则：
E(Z) = (1/σ)E(X) - μ/σ = μ/σ - μ/σ = 0
Var(Z) = (1/σ)² Var(X) = σ² / σ² = 1
This demonstrates the power of E(aX+b) and Var(aX+b). Every time we standardise a variable, we rely on these linear transformation properties implicitly. Similarly, converting Fahrenheit to Celsius or adjusting for systematic measurement errors relies on the same logic.
这展示了 E(aX+b) 和 Var(aX+b) 的力量。每次我们对变量进行标准化，都隐含地依赖这些线性变换性质。类似地，华氏度转摄氏度，或纠正系统测量误差，都基于相同的逻辑。
10. Common Misconceptions and Pitfalls | 常见误区与陷阱
+ b) = aE(X) without adding b. Always check that if a = 1, the formula reduces correctly.
学生们经常混淆期望和方差的规则。常见错误是写 Var(aX + b) = aVar(X) + b，或忘记 a 需要平方。另一个错误是设 E(aX + b) = aE(X) 而漏掉加 b。务必检查当 a = 1 时公式是否能正确简化。
- Misconception 1: "Adding a constant increases variance." No – variance is translation-invariant.
误区1：“加常数会增大方差。”错误——方差具有平移不变性。 - Misconception 2: "Var(aX) = a Var(X)". The correct is a²Var(X).
误区2：“Var(aX) = a Var(X)”。正确形式是 a²Var(X)。 - Misconception 3: "E(aX + b) = aE(X) + b applies only to normal distributions." It applies to any distribution with a defined expectation.
误区3：“E(aX + b) = aE(X) + b 只适用于正态分布。”它适用于任何有定义期望的分布。
Becoming comfortable with distinguishing these cases is essential for success in IB examinations and internal assessments.
熟悉并区分这些情况，对于 IB 考试和内部评估的成功至关重要。
11. Derivation for Discrete Random Variables | 离散随机变量的推导
lues x₁, x₂, ..., xₙ with probabilities p₁, p₂, ..., pₙ. The expectation of Y = aX + b is:
IB 学生通常会先接触离散概率分布。设 X 取值为 x₁, x₂, ..., xₙ，对应概率为 p₁, p₂, ..., pₙ。Y = aX + b 的期望为：
E(Y) = Σ (axᵢ + b)pᵢ = a Σ xᵢpᵢ + b Σ pᵢ = aE(X) + b × 1
For variance, start with the definition Var(Y) = Σ (yᵢ – te yᵢ = axᵢ + b and E(Y) = aE(X) + b. The b terms cancel inside the bracket, leaving (a(xᵢ – E(X)))², so the final sum is a² Σ (xᵢ – ))² pᵢ = a²Var(X). This explicit expansion reinforces why b vanishes and why a is squared.
对于方差，从定义 Var(Y) = Σ (yᵢ – E(Y))² pᵢ 开始。代入 yᵢ = axᵢ + b 以及 E(Y) = aE(X) + b。括号内的 b 项抵消，留下 (a(xᵢ – E(X)))²，最终求和为 a² Σ (xᵢ – E(X))² pᵢ = a²Var(X)。这一展开清楚地说明了为何 b 消失、为何 a 要平方。
a = 2 and b = 3, say X with distribution P(X=1)=0.5, P(X=2)=0.5, gives concrete practice.
通过一个小型数值表格，比如 a = 2, b = 3，设 X 的分布为 P(X=1)=0.5, P(X=2)=0.5，进行具体练习会很有帮助。
12. Summary and Key Takeaways | 总结与要點
To conclude, for any random variable X and constants a, b, the following formulas always hold:
总之，对于任意随机变量 X 及常数 a, b，以下公式始终成立：
- Expected value: E(aX + b) = aE(X) + b
期望值：E(aX + b) = aE(X) + b - Variance: Var(aX + b) = a²Var(X)
方差：Var(aX + b) = a²Var(X) - Standard deviation: σ(aX + b) = |a|σ(X)
标准差：σ(aX + b) = |a|σ(X)
These linear transformation properties are among the most frequently used tools in statistical modelling, data analysis, and probability theory. They enable us to work effortlessly with any change of scale or origin, making it unnecessary to recompute entire distributions from scratch.
这些线性变换性质是统计建模、数据分析和概率论中最常用的工具之一。它们使我们能够轻松应对任何尺度或原点的变化，无需从零开始重新计算完整的分布。
Mastery of this investigation paves the way for understanding more advanced topics like linear combinations of random variables, portfolio theory in finance, and the central limit theorem.
掌握本探究将为理解更高级的主题铺平道路，例如随机变量的线性组合、金融中的投资组合理论以及中心极限定理。
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