Variance of a Discrete Random Variable | 离散随机变量的方差

📚 Variance of a Discrete Random Variable | 离散随机变量的方差

In Edexcel A-Level Mathematics, the variance of a discrete random variable measures how spread out the values of the variable are around the expected value. A small variance means the possible values are tightly clustered around the mean, while a large variance means they are more widely dispersed.

在 Edexcel A-Level 数学中，离散随机变量的方差衡量该变量的取值在期望值周围的分散程度。方差小意味着可能取值紧密聚集在均值附近，而方差大意味着取值更加分散。

---

1. What Is a Discrete Random Variable? | 什么是离散随机变量？

A discrete random variable X is a variable whose numerical value is determined by chance and can only take a finite or countable set of values. Each possible value xᵢ has an associated probability P(X = xᵢ), and the sum of all probabilities must equal 1.

离散随机变量 X 是一个由随机因素决定数值的变量，只能取有限或可数的数值集合。每个可能取值 xᵢ 都有对应的概率 P(X = xᵢ)，所有概率之和必须等于 1。

For example, if X is the score when a fair six-sided die is rolled, then X can take the values 1, 2, 3, 4, 5, 6, each with probability 1/6. Before calculating variance, you should always confirm that the probabilities form a valid distribution.

例如，若 X 表示掷一枚公平六面骰子的得分，那么 X 可以取 1、2、3、4、5、6，每个值的概率都是 1/6。在计算方差之前，你应始终确认这些概率构成一个有效的分布。

---

2. Expected Value E(X) as a Starting Point | 期望值 E(X) 作为起点

The variance is based on the expected value, so you must be confident with E(X) first. For a discrete random variable X, the expected value is the long-run average value of X if the experiment is repeated many times.

方差以期望值为基础，因此你必须首先熟练掌握 E(X)。对于离散随机变量 X，期望值是当试验重复很多次时 X 的长期平均值。

The expected value is calculated by multiplying each possible value by its probability and then summing the results:

期望值的计算方法是将每个可能取值乘以其对应概率，然后求和：

E(X) = Σ xᵢ pᵢ

Here pᵢ means P(X = xᵢ). The expected value is often denoted by the Greek letter μ, so μ = E(X). It is a measure of central location, not a measure of spread.

这里的 pᵢ 表示 P(X = xᵢ)。期望值通常用希腊字母 μ 表示，即 μ = E(X)。它是位置中心的度量，而不是分散程度的度量。

---

3. Defining Variance and Standard Deviation | 方差与标准差的定义

The variance of X measures the average squared distance of the values from the mean μ. Squaring ensures that negative and positive deviations do not cancel each other out.

X 的方差衡量各取值与均值 μ 之间距离平方的平均值。平方可以确保负偏差和正偏差不会相互抵消。

The formal definition of variance is:

方差的正式定义是：

Var(X) = E[(X − μ)²] = Σ (xᵢ − μ)² pᵢ

This formula says: take each possible value xᵢ, subtract the mean μ, square the result, multiply by the probability pᵢ, and then add all these terms together.

这个公式表示：取每个可能取值 xᵢ，减去均值 μ，将结果平方，再乘以其概率 pᵢ，然后把所有这些项加起来。

The standard deviation, denoted by σ, is the positive square root of the variance:

标准差用 σ 表示，是方差的正平方根：

σ = √Var(X)

Standard deviation is often more useful in interpretation because it has the same units as the original data.

标准差在解释时通常更有用，因为它的单位与原始数据相同。

---

4. The Faster Variance Formula: Var(X) = E(X²) − [E(X)]² | 快捷方差公式：Var(X) = E(X²) − [E(X)]²

Although the definition is important, Edexcel exam solutions almost always use a faster computational formula. Expanding the definition and using Σ pᵢ = 1 gives:

虽然定义很重要，但 Edexcel 考试解题几乎总是使用一个更快的计算公式。展开定义并利用 Σ pᵢ = 1 可以得到：

Var(X) = Σ (xᵢ − μ)² pᵢ = Σ xᵢ² pᵢ − 2μ Σ xᵢ pᵢ + μ² Σ pᵢ

Since Σ xᵢ pᵢ = μ and Σ pᵢ = 1, this simplifies to:

由于 Σ xᵢ pᵢ = μ 且 Σ pᵢ = 1，上式可简化为：

Var(X) = E(X²) − μ² = E(X²) − [E(X)]²

E(X²) is found by squaring each value first, then multiplying by its probability and summing: E(X²) = Σ xᵢ² pᵢ. This is not the same as [E(X)]².

E(X²) 的计算方法是先将每个取值平方，然后乘以其对应概率并求和：E(X²) = Σ xᵢ² pᵢ。这与 [E(X)]² 不同。

In practice, set up a table with columns for x, P(X = x), x pᵢ and x² pᵢ. This reduces arithmetic errors and shows clear working.

在实际计算中，可以建立一个表格，列出 x、P(X = x)、x pᵢ 和 x² pᵢ 四列。这样可以减少计算错误，并使解题过程更清晰。

---

5. Worked Example: Rolling a Fair Die | 例题：掷一枚公平骰子

Let X be the score when a fair six-sided die is rolled. Each value from 1 to 6 has probability 1/6. We first construct a probability distribution table.

设 X 为掷一枚公平六面骰子的得分。1 到 6 的每个值的概率都是 1/6。我们首先构建一个概率分布表。

x	1	2	3	4	5	6
P(X = x)	1/6	1/6	1/6	1/6	1/6	1/6

The expected value is:

期望值为：

E(X) = (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 21/6 = 3.5

Next calculate E(X²) by squaring each value first:

接下来计算 E(X²)，先将每个值平方：

E(X²) = (1² + 2² + 3² + 4² + 5² + 6²) ÷ 6 = (1 + 4 + 9 + 16 + 25 + 36) ÷ 6 = 91/6

Therefore the variance is:

因此方差为：

Var(X) = E(X²) − [E(X)]² = 91/6 − (7/2)² = 91/6 − 49/4

Var(X) = 182/12 − 147/12 = 35/12 ≈ 2.9167

The standard deviation is σ = √(35/12) ≈ 1.708. This tells us that, on average, the score is about 1.71 units away from the mean of 3.5.

标准差为 σ = √(35/12) ≈ 1.708。这告诉我们，平均而言，得分与均值 3.5 的距离约为 1.71 个单位。

---

6. Linear Transformations: Var(aX + b) | 线性变换：Var(aX + b)

Edexcel frequently tests how variance changes when a random variable is transformed linearly. If a and b are constants, then:

Edexcel 经常考查当随机变量进行线性变换时方差如何变化。如果 a 和 b 是常数，那么：

E(aX + b) = aE(X) + b

Var(aX + b) = a² Var(X)

The constant a multiplies the spread, so the variance is multiplied by a². The constant b only shifts all values left or right, so it does not change the spread at all.

常数 a 会成倍改变分散程度，因此方差要乘以 a²。常数 b 只是将所有值向左或向右平移，所以完全不会改变分散程度。

For example, if Y = 2X + 3 and Var(X) = 5, then Var(Y) = 2² × 5 = 20. If Z = −3X + 1, then Var(Z) = (−3)² × Var(X) = 9 Var(X).

例如，如果 Y = 2X + 3 且 Var(X) = 5，则 Var(Y) = 2² × 5 = 20。如果 Z = −3X + 1，则 Var(Z) = (−3)² × Var(X) = 9 Var(X)。

---

7. Why the Constant b Does Not Affect Variance | 为什么常数 b 不影响方差

Adding a constant b shifts the whole distribution along the number line but keeps the gaps between values unchanged. Since variance measures spread, not location, it must remain the same under translation.

加上常数 b 会将整个分布沿数轴平移，但各取值之间的间隔保持不变。由于方差衡量的是分散程度而不是位置，因此在平移下方差必须保持不变。

You can see this from the definition: if Y = X + b, then μᵧ = μₓ + b, so each deviation becomes (xᵢ + b) − (μₓ + b) = xᵢ − μₓ. The deviations are unchanged, so the variance is unchanged.

从定义可以看出：如果 Y = X + b，则 μᵧ = μₓ + b，因此每个偏差变为 (xᵢ + b) − (μₓ + b) = xᵢ − μₓ。偏差不变，所以方差也不变。

A common application is changing temperature from degrees Celsius to degrees Fahrenheit using F = 1.8C + 32. The variance of F is 1.8² times the variance of C, and the 32 has no effect on variance.

一个常见的应用是使用 F = 1.8C + 32 将温度从摄氏度转换为华氏度。F 的方差是 C 的方差的 1.8² 倍，而 32 对方差没有影响。

---

8. Standard Deviation: Returning to Original Units | 标准差：回到原始单位

Variance is measured in squared units. If X is measured in centimetres, then Var(X) is measured in cm², which can be difficult to interpret directly.

方差的单位是原始单位的平方。如果 X 以厘米为单位，那么 Var(X) 的单位是 cm²，这通常难以直接解释。

The standard deviation σ = √Var(X) returns the measure of spread to the original units. If Var(X) = 35/12 cm² for the die score example, then σ ≈ 1.708 cm.

标准差 σ = √Var(X) 将分散程度的度量恢复为原始单位。如果骰子得分例子中 Var(X) = 35/12 cm²，那么 σ ≈ 1.708 cm。

When a linear transformation is applied, the standard deviation changes by multiplying by the absolute value of a:

当进行线性变换时，标准差的变化是乘以 a 的绝对值：

σ(aX + b) = |a| σ(X)

This is because standard deviation is always non-negative.

这是因为标准差始终为非负值。

---

9. Common Mistakes in Variance Calculations | 方差计算中的常见错误

One common mistake is confusing E(X²) with [E(X)]². E(X²) means square each value first, then take the expectation. [E(X)]² means take the expectation first, then square the result. These are generally different.

一个常见错误是混淆 E(X²) 和 [E(X)]²。E(X²) 表示先将每个值平方，再求期望。[E(X)]² 表示先求期望，再将结果平方。这两者通常不同。

Another frequent error is forgetting to square the values when calculating E(X²). Always use a table with an x² pᵢ column to avoid this mistake.

另一个常见错误是在计算 E(X²) 时忘记将取值平方。一定要使用带有 x² pᵢ 列的表格来避免这个错误。

Students also sometimes multiply variance by a rather than a² under a linear transformation. Remember that Var(aX + b) = a² Var(X), not a Var(X).

学生在进行线性变换时，有时会把方差乘以 a 而不是 a²。请记住 Var(aX + b) = a² Var(X)，而不是 a Var(X)。

Finally, check that your variance is non-negative. A negative variance is impossible, so it indicates an arithmetic mistake earlier in the working.

最后，检查方差是否为非负值。负方差是不可能的，所以出现负值说明前面的计算存在算术错误。

---

10. Exam-Style Tips and Interpretation | 考试技巧与结果解读

In Edexcel exam questions, you should always show the full probability distribution table before calculating E(X) and E(X²). Clear tabular working is expected and can earn method marks even if the final answer is wrong.

在 Edexcel 考试题中，你应在计算 E(X) 和 E(X²) 之前先展示完整的概率分布表。清晰的表格计算过程是评分所期望的，即使最终答案错误也可能获得方法分。

When asked to interpret variance or standard deviation, say that a larger value means the data is more spread out around the mean, while a smaller value means the data is more consistent.

当题目要求解释方差或标准差时，可以说较大的值意味着数据在均值周围更分散，而较小的值意味着数据更集中、更一致。

For questions involving an unknown probability, use the fact that the probabilities sum to 1, and sometimes use a given expected value to set up equations. Once you find the distribution, apply the standard variance formula.

对于涉及未知概率的题目，可以利用概率之和为 1 这一事实，有时还可以利用给定的期望值建立方程。求出分布后，再套用标准方差公式。

Keep fractions and surds in exact form unless the question asks for a decimal approximation. This shows accuracy and avoids rounding errors in subsequent parts.

除非题目要求给出小数近似值，否则应保留分数和根式的精确形式。这可以体现准确性，并避免后续计算中的舍入误差。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）