📚 Properties of Expectation and Variance | 期望与方差的运算性质
Expectation and variance are two fundamental measures that summarise a probability distribution: expectation describes the centre (long-run average), while variance describes the spread around the centre. In IB Mathematics, mastering their algebraic properties allows us to compute these quantities efficiently without recalculating the full distribution for every linear transformation.
期望（Expectation）和方差（Variance）是概率分布的两个基本度量：期望描述分布的中心位置（长期平均值），方差描述数据围绕中心波动的程度。在 IB 数学中，熟练掌握它们的代数运算性质，可以帮助我们在面对线性变换或新组合变量时，无需重新计算完整分布即可快速求解。
1. Definition of Expectation | 期望的定义
For a discrete random variable X with probability function P(X = x), the expectation is defined as the weighted sum of all possible values.
对于离散型随机变量 X，若其概率函数为 P(X = x)，则期望定义为所有可能取值的加权和。
E(X) = Σ x·P(X = x)
For a continuous random variable with probability density function f(x), the expectation is the integral of x·f(x).
对于连续型随机变量，若其概率密度函数为 f(x)，则期望是 x·f(x) 的积分。
E(X) = ∫ x·f(x) dx
2. Linearity of Expectation | 期望的线性性质
The expectation operator is linear. For any constants a and b, and any random variable X:
期望算子具有线性性质。对任意常数 a、b 以及任意随机变量 X：
E(aX + b) = a·E(X) + b
Moreover, for two random variables X and Y, the expectation of their sum is the sum of their expectations:
此外，对于两个随机变量 X 和 Y，它们之和的期望等于各自期望之和：
E(X + Y) = E(X) + E(Y)
This property does not require X and Y to be independent. It holds universally because the expectation is a linear combination of probabilities.
该性质不要求 X 与 Y 独立。因为期望本质上是概率的线性组合，所以它总是成立。
3. Expectation of a Product | 乘积的期望
If X and Y are independent random variables, then the expectation of their product is the product of their expectations:
如果 X 与 Y 是独立随机变量，那么它们乘积的期望等于各自期望的乘积：
E(XY) = E(X)·E(Y) (if X, Y independent)
Be careful: the converse is not true. Knowing E(XY) = E(X)E(Y) does not imply X and Y are independent — it only means they are uncorrelated.
注意：逆命题不成立。已知 E(XY) = E(X)E(Y) 并不能推出 X 与 Y 独立，只能说明它们不相关。
4. Definition of Variance | 方差的定义
Variance measures the average squared deviation from the mean μ = E(X).
方差度量随机变量与其均值 μ = E(X) 之间的平均平方偏差。
Var(X) = E[(X − μ)²]
For a discrete distribution, this becomes Var(X) = Σ (x − μ)²·P(X = x). For a continuous distribution, it is the integral ∫ (x − μ)² f(x) dx.
对离散分布，方差为 Var(X) = Σ (x − μ)²·P(X = x)。对连续分布，方差为积分 ∫ (x − μ)² f(x) dx。
5. Computing Variance via E(X²) | 用 E(X²) 计算方差
A very useful formula avoids expanding each deviation:
一个非常有用的公式可以避免逐一展开离差平方：
Var(X) = E(X²) − [E(X)]²
Since E(X) is often easy to find, this formula is recommended when the distribution table of X² is available. Remember: E(X²) is generally not equal to [E(X)]².
由于 E(X) 通常容易求得，当 X² 的分布表可用时，推荐使用这个公式。记住：E(X²) 通常不等于 [E(X)]²。
6. Variance under Linear Transformation | 线性变换下的方差
If we apply a linear transformation Y = aX + b, the scale factor a changes the spread, but the shift b does not affect spread:
若对随机变量进行线性变换 Y = aX + b，比例系数 a 会改变波动幅度，而平移量 b 不会影响离散程度：
Var(aX + b) = a²·Var(X)
For example, Var(2X) = 4Var(X), and Var(X + 5) = Var(X). The constant b disappears completely.
例如，Var(2X) = 4Var(X)，而 Var(X + 5) = Var(X)。常数 b 被完全消去。
7. Variance of the Sum of Independent Variables | 独立变量之和的方差
If X and Y are independent, then the variance of their sum is the sum of their variances:
如果 X 与 Y 独立，则它们之和的方差等于各自方差之和：
Var(X + Y) = Var(X) + Var(Y)
For a difference: Var(X − Y) = Var(X) + Var(Y) as well, because subtracting Y reverses its sign but squares it.
对于之差：Var(X − Y) = Var(X) + Var(Y) 同样成立，因为减去 Y 相当于乘上 −1，而平方后不变。
If independence is not assumed, the general formula involves the covariance: Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y).
若不能保证独立，一般公式需要引入协方差：Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y)。
8. Common Discrete Distributions | 常见离散分布
The table below summarises the expectation and variance for the most common discrete distributions in IB.
下表总结了 IB 中最常见离散分布的期望与方差。
| Distribution | E(X) | Var(X) |
| Bernoulli(p) | p | p(1 − p) |
| Binomial(n, p) | np | np(1 − p) |
| Poisson(λ) | λ | λ |
Using these standard results saves time: for example, if X ~ B(10, 0.3), then E(X) = 3 and Var(X) = 10 × 0.3 × 0.7 = 2.1.
利用这些标准结果可以节省时间：例如，若 X ~ B(10, 0.3)，则 E(X) = 3，Var(X) = 10 × 0.3 × 0.7 = 2.1。
9. Common Continuous Distributions | 常见连续分布
For continuous models, the same ideas apply.
对于连续型分布，上述思想同样适用。
| Distribution | E(X) | Var(X) |
| Uniform(a, b) | (a + b)/2 | (b − a)²/12 |
| Normal(μ, σ²) | μ | σ² |
| Exponential(λ) | 1/λ | 1/λ² |
For an exponential distribution with rate λ, the variance is quite large relative to the mean, reflecting its right-skewed shape.
对于参数为 λ 的指数分布，方差相对均值较大，这反映了其右偏的形状。
10. Standardisation | 标准化变量
A common linear transformation is the standardisation of a random variable:
一种常见的线性变换是随机变量的标准化：
Z = (X − μ) / σ
Using the linear properties, we obtain:
利用线性性质，我们可以得到：
E(Z) = 0, Var(Z) = 1
This creates a variable with mean 0 and variance 1, which is essential for the standard normal distribution and for comparing variables with different units.
这构造了一个均值为 0、方差为 1 的变量，对于标准正态分布以及比较不同单位的变量非常关键。
11. Common Pitfalls and Worked Examples | 常见陷阱与例题
Pitfall 1: Confusing E(X²) with [E(X)]². For example, if X takes values 0 and 2 with equal probability, then E(X) = 1, [E(X)]² = 1, but E(X²) = (0 + 4)/2 = 2, so Var(X) = 2 − 1 = 1.
陷阱一：混淆 E(X²) 与 [E(X)]²。例如，若 X 等概率取 0 和 2，则 E(X) = 1，[E(X)]² = 1，但 E(X²) = (0 + 4)/2 = 2，所以 Var(X) = 2 − 1 = 1。
Pitfall 2: Forgetting to square the constant factor. Var(3X) = 9Var(X), not 3Var(X).
陷阱二：忘记平方常数因子。Var(3X) = 9Var(X)，而不是 3Var(X)。
Worked example: If E(X) = 2, Var(X) = 5, find E(4X + 3) and Var(4X + 3).
例题：若 E(X) = 2，Var(X) = 5，求 E(4X + 3) 与 Var(4X + 3)。
E(4X + 3) = 4×2 + 3 = 11, Var(4X + 3) = 4²×5 = 80
Another example: X and Y are independent with E(X) = 1, Var(X) = 2, E(Y) = 3, Var(Y) = 4. Find E(2X − Y) and Var(2X − Y).
另一例：X 与 Y 独立，E(X) = 1，Var(X) = 2，E(Y) = 3，Var(Y) = 4。求 E(2X − Y) 与 Var(2X − Y)。
E(2X − Y) = 2×1 − 3 = −1, Var(2X − Y) = 4×2 + 4 = 12
12. Summary | 总结
The expectation is a linear operator: E(aX + bY) = aE(X) + bE(Y). The variance, however, only respects addition under independence, and its scale factor is squared: Var(aX + b) = a²Var(X).
期望是线性算子：E(aX + bY) = aE(X) + bE(Y)。而方差只在独立条件下可加，且尺度因子需要平方：Var(aX + b) = a²Var(X)。
Always compute E(X²) separately from [E(X)]², and use the standard formulas for named distributions to save time in the IB exam.
务必单独计算 E(X²)，不要与 [E(X)]² 混淆；同时利用常见分布的公式，可以在 IB 考试中节省大量时间。
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