📚 Mean and Variance of a Distribution | 分布的均值与方差
For any random variable X, the mean and variance summarise the centre and spread of its probability distribution. In Edexcel A Level Mathematics, you need to calculate E(X) and Var(X) for discrete distributions, apply linear transformations, and use standard results for distributions such as the binomial.
对于任意随机变量 X，均值和方差概括了其概率分布的中心位置与离散程度。在 Edexcel A Level 数学中，你需要能够计算离散分布的 E(X) 和 Var(X)，应用线性变换，并使用二项分布等常见分布的标准结果。
1. Probability Distributions and Notation | 概率分布与记号
A discrete random variable X takes values x₁, x₂, … with probabilities P(X=xᵢ). The probabilities must satisfy 0 ≤ P(X=xᵢ) ≤ 1 and Σ P(X=xᵢ) = 1.
离散随机变量 X 取值为 x₁、x₂、…，对应概率为 P(X=xᵢ)。概率必须满足 0 ≤ P(X=xᵢ) ≤ 1 且 Σ P(X=xᵢ) = 1。
The mean is the long-run average value if the experiment is repeated many times. It is a measure of location.
均值是多次重复试验时随机变量的长期平均值，它是位置的度量。
The variance measures how far the values of X are spread from the mean. A small variance means values are tightly clustered around the mean.
方差衡量 X 的取值偏离均值的程度。方差很小说明取值紧密围绕均值。
2. Expected Value E(X) | 期望值 E(X)
The expected value of a discrete random variable X is defined as
离散随机变量 X 的期望值定义为
E(X) = Σ x P(X=x)
where the sum is taken over all possible values of X. This is a probability-weighted average.
其中求和遍及 X 的所有可能取值。这是以概率为权重的加权平均。
Example: if X takes values 1, 2, 3 with probabilities 0.2, 0.5, 0.3, then E(X) = 1(0.2) + 2(0.5) + 3(0.3) = 2.1.
例如：若 X 取值 1、2、3，概率分别为 0.2、0.5、0.3，则 E(X) = 1(0.2) + 2(0.5) + 3(0.3) = 2.1。
Some textbooks use μ to denote E(X). Both μ and E(X) mean the same thing.
有些教材用 μ 表示 E(X)。μ 和 E(X) 含义相同。
3. Variance Var(X) | 方差 Var(X)
The variance of X is the expected squared deviation from the mean:
X 的方差是偏离均值的平方的期望：
Var(X) = E[(X – μ)²] = Σ (x – μ)² P(X=x)
It cannot be negative. A variance of zero means X has only one possible value with probability 1.
方差不能为负。方差为零意味着 X 只有一个可能取值，其概率为 1。
For the previous example with μ = 2.1, Var(X) = (1-2.1)²(0.2) + (2-2.1)²(0.5) + (3-2.1)²(0.3) = 0.49.
在前例中 μ = 2.1，Var(X) = (1-2.1)²(0.2) + (2-2.1)²(0.5) + (3-2.1)²(0.3) = 0.49。
Because the units of variance are the square of the units of X, the standard deviation is often easier to interpret.
由于方差的单位是 X 单位的平方，标准差通常更容易解释。
4. Alternative Variance Formula | 方差的计算公式
The most efficient way to calculate Var(X) in exam questions is often:
考试中计算 Var(X) 最常用的高效公式是：
Var(X) = E(X²) – [E(X)]²
where E(X²) = Σ x² P(X=x). This avoids subtracting the mean from every value separately.
其中 E(X²) = Σ x² P(X=x)。这样就不必对每个取值分别减去均值。
For the example above, E(X²) = 1²(0.2) + 2²(0.5) + 3²(0.3) = 4.9, so Var(X) = 4.9 – 2.1² = 4.9 – 4.41 = 0.49.
在上例中，E(X²) = 1²(0.2) + 2²(0.5) + 3²(0.3) = 4.9，因此 Var(X) = 4.9 – 2.1² = 4.9 – 4.41 = 0.49。
This formula is algebraic, so it is valid only when E(X²) is finite.
该公式是代数恒等式，因此在 E(X²) 有限时始终成立。
5. Linear Transformations | 线性变换
If Y = aX + b, where a and b are constants, then the mean and variance transform as follows:
如果 Y = aX + b，其中 a 和 b 是常数，则均值和方差的变换规则如下：
E(aX + b) = aE(X) + b
Var(aX + b) = a² Var(X)
Adding the constant b shifts the mean but does not change the spread, so it does not affect the variance.
加上常数 b 会平移均值，但不改变离散程度，因此它不影响方差。
Multiplying by a scales both the mean and the standard deviation by a factor of |a|. The variance is scaled by a².
乘以 a 会使均值和标准差都按 |a| 的倍数缩放，而方差按 a² 缩放。
For example, if E(X)=5 and Var(X)=4, then for Y=3X-2: E(Y)=3(5)-2=13 and Var(Y)=3²(4)=36.
例如，若 E(X)=5、Var(X)=4，则对 Y=3X-2：E(Y)=3(5)-2=13，且 Var(Y)=3²(4)=36。
6. Standard Deviation | 标准差
The standard deviation is the positive square root of the variance:
标准差是方差的正平方根：
σ = √Var(X)
It has the same units as X, so it is often used to describe spread in context.
它与 X 的单位相同，因此常在具体情境中用于描述离散程度。
For example, if Var(X) = 0.49, then σ = √0.49 = 0.7.
例如，若 Var(X) = 0.49，则 σ = √0.49 = 0.7。
Do not confuse variance with standard deviation: Edexcel questions often ask for one and then the other.
不要混淆方差和标准差：Edexcel 题目经常要求先求一个，再求另一个。
7. Discrete Uniform Distribution | 离散均匀分布
A discrete uniform distribution on the values 1, 2, …, n gives each value probability 1/n.
取值 1、2、…、n 的离散均匀分布给每个值的概率都是 1/n。
The mean and variance are:
其均值和方差为：
E(X) = (n + 1) / 2
Var(X) = (n² – 1) / 12
For example, when rolling a fair six-sided die, X ~ Uniform(1,6), so E(X) = 7/2 = 3.5 and Var(X) = (36 – 1)/12 = 35/12.
例如，掷一枚公平的六面骰子时，X ~ Uniform(1,6)，所以 E(X) = 7/2 = 3.5，且 Var(X) = (36 – 1)/12 = 35/12。
These standard results can save time, but you must be able to derive them from the definitions if required.
这些标准结果可以节省时间，但你仍需能够根据定义进行推导。
8. Binomial Distribution Mean and Variance | 二项分布的均值与方差
If X ~ B(n, p), where n is the number of independent trials and p is the probability of success in each trial, then:
如果 X ~ B(n, p)，其中 n 是独立试验次数，p 是每次试验成功的概率，则：
E(X) = np
Var(X) = np(1 – p)
These are given in the Edexcel formula booklet, but you should understand what they mean.
这些公式在 Edexcel 公式手册中给出，但你应该理解它们的含义。
If X ~ B(20, 0.3), then E(X) = 20(0.3) = 6 and Var(X) = 20(0.3)(0.7) = 4.2.
若 X ~ B(20, 0.3)，则 E(X) = 20(0.3) = 6，且 Var(X) = 20(0.3)(0.7) = 4.2。
Notice that the variance is largest when p = 0.5, because np(1 – p) is maximised at p = 0.5 for a fixed n.
注意，当 p = 0.5 时方差最大，因为对于固定的 n，np(1 – p) 在 p = 0.5 处取得最大值。
9. Worked Example | 例题解析
A discrete random variable X has the following probability distribution:
离散随机变量 X 的概率分布如下：
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