📚 Mean and Variance of a Distribution | 分布的均值与方差
In A‑Level Statistics, the mean and variance of a probability distribution are the two most fundamental measures that summarise a random variable’s behaviour. The mean tells you where the centre of the distribution lies, while the variance (or its square root, the standard deviation) quantifies how spread out the values are around that centre. Mastering the calculation and properties of expectation and variance is essential for tackling everything from discrete probability distributions to hypothesis testing and modelling real‑world data.
在A‑Level统计学中，概率分布的均值和方差是概括随机变量行为的两个最基本度量。均值告诉你分布的中心位置，而方差（或其平方根——标准差）则量化了数值围绕中心的离散程度。掌握期望和方差的计算及性质，对于从离散概率分布到假设检验、建立现实世界数据模型等各个环节都至关重要。
1. What is a Random Variable? | 什么是随机变量？
e random variable can only take a countable number of distinct values, each with an associated probability. The list of all possible values together with their probabilities is called the probability distribution of X. For A‑ ostly work with discrete random variables defined by a probability mass function P(X = x).
随机变量是其取值依赖于随机试验结果的变量。我们通常用大写字母（如X）表示它。离散随机变量只能取可数个不同的值，每个值对应一个概率。所有可能取值及其概率的列表称为X的概率分布。在Edexcel A‑Level考试中，你主要处理由概率质量函数P(X = x)定义的离散随机变量。
2. Expectation (Mean) of a Discrete Random Variable | 离散随机变量的期望（均值）
The expected value, often called the mean and written as E(X) or μ, is the long‑ iable X taking values xᵢ with probabilities pᵢ = P(X = xᵢ), the expectation is defined as:
E(X) = Σ xᵢ pᵢ = Σ x · P(X = x)
期望值，常称为均值，写作E(X)或μ，是如果试验重复多次时随机变量的长期平均值。对于取值为xᵢ、相应概率为pᵢ = P(X = xᵢ)的离散随机变量X，期望定义为：
E(X) = Σ xᵢ pᵢ = Σ x · P(X = x)
3. Calculating E(X) – Worked Example | 计算E(X) – 实例
Suppose a fair six‑ mber shown. The probability distribution is P(X = x) = 1/6 for x = 1, 2, 3, 4, 5, 6. Then:
E(X) = 1×(1/6) + 2×(1/6) + 3×(1/6) + 4×(1/6) + 5×(1/6) + 6×(1/6) = 3.5
Notice that 3.5 is not a possible outcome of a single roll, but it represents the average over many rolls.
假设掷一个公平的六面骰子，随机变量X为显示的点数。概率分布为P(X = x) = 1/6，x = 1, 2, 3, 4, 5, 6。则：
E(X) = 1×(1/6) + 2×(1/6) + 3×(1/6) + 4×(1/6) + 5×(1/6) + 6×(1/6) = 3.5
注意，3.5并不是单次掷骰子可能出现的结果，但它表示多次掷骰子的平均值。
4. Variance of a Discrete Random Variable | 离散随机变量的方差
ed deviation from the mean. It is denoted by Var(X) or σ². The definition is Var(X) = E[(X – μ)²], but a much more convenient formula for calculation is:
Var(X) = E(X²) – [E(X)]²
= Σ x² P(X = x). You must always calculate E(X²) first, then square the mean and subtract.
方差衡量的是各个取值与均值之差的平方的平均值。它记为Var(X)或σ²。定义是Var(X) = E[(X – μ)²]，但一个更方便的计算公式是：
Var(X) = E(X²) – [E(X)]²
这里E(X²) = Σ x² P(X = x)。你必须先计算E(X²)，再减去均值的平方。
5. Calculating Var(X) – Worked Example | 计算Var(X) – 实例
For the die example, we first compute E(X²):
E(X²) = 1²×(1/6) + 2²×(1/6) + 3²×(1/6) + 4²×(1/6) + 5²×(1/6) + 6²×(1/6) = 91/6 ≈ 15.1667
Then Var(X) = 91/6 – (3.5)² = 91/6 – 12.25 = 91/6 – 49/4 = (182 – 147)/12 = 35/12 ≈ 2.917.
所以，这枚骰子点数的方差约为2.917。
对于骰子的例子，我们先计算E(X²)：
E(X²) = 1²×(1/6) + 2²×(1/6) + 3²×(1/6) + 4²×(1/6) + 5²×(1/6) + 6²×(1/6) = 91/6 ≈ 15.1667
然后Var(X) = 91/6 – (3.5)² = 91/6 – 12.25 = 91/6 – 49/4 = (182 – 147)/12 = 35/12 ≈ 2.917。
6. Standard Deviation | 标准差
as the original data, making it easier to interpret than variance. In the die example, σ = √35/12 ≈ 1.71.
标准差σ就是方差的正平方根：σ = √Var(X)。它的单位与原始数据相同，因此比方差更容易解释。在骰子的例子中，σ = √35/12 ≈ 1.71。
7. Linear Transformations of a Random Variable | 随机变量的线性变换
iable is Y = aX + b. The mean and variance of Y are related to those of X in a simple way:
E(aX + b) = a E(X) + b
Var(aX + b) = a² Var(X)
Notice that adding a constant shifts the mean but does not affect the spread, so it adds no variance. Multiplication by a scales both the mean and the variance, but the variance is multiplied by a².
如果你将X乘以一个常数a，再加上一个常数b，得到新的随机变量Y = aX + b。Y的均值和方差与X的均值和方差有如下简单关系：
E(aX + b) = a E(X) + b
Var(aX + b) = a² Var(X)
注意，加上一个常数会平移均值，但不影响离散程度，因此不会增加方差。乘以一个常数会同时缩放均值和方差，但方差要乘以a²。
8. Mean and Variance of aX + b – Worked Example | aX+b的均值和方差 – 实例
A random variable X has E(X) = 5 and Var(X) = 4. Find the mean and variance of Y = 3X – 2.
E(Y) = 3 × 5 – 2 = 13. Var(Y) = 3² × 4 = 9 × 4 = 36. The standard deviation of Y is √36 = 6.
useful in coding and in standardising variables to the standard normal distribution Z = (X – μ)/σ.
已知随机变量X的E(X) = 5，Var(X) = 4。求Y = 3X – 2的均值和方差。
E(Y) = 3 × 5 – 2 = 13。Var(Y) = 3² × 4 = 9 × 4 = 36。Y的标准差为√36 = 6。
这些变换规则在编码以及将变量标准化为标准正态分布Z = (X – μ)/σ时非常有用。
9. Combining Independent Random Variables | 独立随机变量的组合
When two random variables X and Y are independent, the mean and variance of their sum or difference follow these rules:
E(X ± Y) = E(X) ± E(Y)
Var(X ± Y) = Var(X) + Var(Y)
Note the variance always adds, regardless of whether you are finding the variance of X + Y or X – Y. For more than two independent variables, the variances simply add. If the variables are not independent, you cannot simply add variances; you would need covariance.
当两个随机变量X和Y相互独立时，它们的和或差的均值和方差遵循以下规则：
E(X ± Y) = E(X) ± E(Y)
Var(X ± Y) = Var(X) + Var(Y)
请注意，方差总是相加，无论你求的是X + Y的方差还是X – Y的方差。对于多于两个独立变量，方差也是简单相加。如果变量不独立，就不能简单相加方差，而需要考虑协方差。
10. Special Distributions: Binomial Mean and Variance | 特殊分布：二项分布的均值与方差
For a binomial distribution X ~ B(n, p), the mean and variance can be derived from the general definitions, but they have simple ready‑made formulas:
E(X) = np
Var(X) = np(1 – p) = npq, where q = 1 – p
20 and p = 0.3, then E(X) = 6 and Var(X) = 20 × 0.3 × 0.7 = 4.2. The standard deviation is √4 saves time and helps you check your answers when using probability tables or calculators.
对于二项分布X ~ B(n, p)，均值和方差可以根据一般定义推导出来，但它们有简单的现成公式：
E(X) = np
Var(X) = np(1 – p) = npq，其中q = 1 – p
例如，若n = 20，p = 0.3，则E(X) = 6，Var(X) = 20 × 0.3 × 0.7 = 4.2。标准差为√4.2 ≈ 2.05。掌握这些快捷公式可以节省时间，并帮助你在使用概率表或计算器时检查答案。
11. Using Summary Statistics Formulae | 使用汇总统计公式
ice to organize your working in a table with columns for x, P(X = x), x·P(X = x) and x²·P(X = x). Then sum the last two columns to obtain E(X) and E(X²). Always state the formula you are using before substituting, e.g., "Using Var(X) = E(X²) – [E(X)]²". This approach helps avoid careless arithmetic errors.
在许多考试题目中，你会得到一个频数表或概率分布，并要求计算均值和方差。一个好的做法是，将计算过程组织成表格，包含x、P(X = x)、x·P(X = x)和x²·P(X = x)这几列。然后对后两列求和，得到E(X)和E(X²)。在代入前，务必先写出所使用的公式，例如，“使用Var(X) = E(X²) – [E(X)]²”。这种方法有助于避免粗心的算术错误。
12. Common Mistakes and Exam Tips | 常见错误与考试提示
- Forgetting to square a in variance: A frequent error is writing Var(aX + b) = a Var(X). Remember, it must be a² Var(X).
- Adding variances for differences: Students sometimes subtract variances for X – Y, but the correct formula is Var(X – Y) = Var(X) + Var(Y).
- Using probabilities that do not sum to 1 e calculating mean and variance. If the probabilities are given as fractions, check the sum.
- Rounding too early: Keep calculations in exact fractions or to a high degree of precision until the final answer.
- Interpreting variance vs standard deviation: In context questions, make sure you use the correct measure – variance is in squared units, standard deviation is in original units.
- 忘记对方差中的系数a平方：一个常见错误是写成Var(aX + b) = a Var(X)。记住，必须是a² Var(X)。
- 对差值做方差减法：学生有时会对X – Y的方差做减法，但正确的公式是Var(X – Y) = Var(X) + Var(Y)。
- 使用的概率和不等于1：在计算均值和方差之前，总是要验证Σ P(X = x) = 1。如果概率以分数形式给出，检查其总和。
- 过早四舍五入：在得到最终答案之前，尽量保持精确分数或高精度小数。
- 混淆方差和标准差的解释：在情境题中，确保使用正确的度量——方差的单位是平方单位，标准差的单位与原数据相同。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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