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 A-Level Mathematics, understanding the variance of a discrete random variable is essential for describing the variability of data. This article will guide you through the definition, calculation, properties, and exam-style applications of variance, following the Edexcel specification.

在A-Level数学中，理解离散随机变量的方差对于描述数据的变异性至关重要。本文将按照Edexcel考试大纲，带你掌握方差的定义、计算、性质以及考试风格的应用。

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

A discrete random variable is a variable that can take a countable number of distinct values, each with an associated probability. These probabilities must sum to 1.

离散随机变量是一个可以取有限个或可数个不同值的变量，每个值都有对应的概率，且所有概率之和为1。

X is often displayed in a table, listing each possible value x and its probability P(X=x).

离散随机变量X的概率分布通常用表格表示，列出每个可能的取值x及其概率P(X=x)。

Understanding the distribution's shape is the first step; variance then quantifies how spread out these values are.

理解分布的形状是第一步；方差则量化这些值的分散程度。

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2. Expected Value (Mean) Revision | 期望值（均值）复习

The expected value of a discrete random variable X, denoted by E(X) or μ, is a measure of central tendency. It is calculated as the sum of each value multiplied by its probability.

离散随机变量X的期望值，记作E(X)或μ，是集中趋势的度量。计算方法是每个取值乘以对应概率的总和。

E(X) = Σ x · P(X=x)

For any probability distribution, E(X) represents the long-run average if the experiment were repeated many times.

对于任何概率分布，E(X)表示如果实验重复多次的长期平均值。

You must be confident in finding E(X) before moving on to variance, as variance is defined using the expected value.

在进入方差之前，必须熟练计算E(X)，因为方差的定义依赖于期望值。

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3. Definition of Variance | 方差的定义

Variance, denoted by Var(X), measures the average squared deviation of the random variable from its mean. It quantifies the spread of the distribution.

方差，记作Var(X)，度量随机变量与其均值的偏差的平方的平均值，量化了分布的离散程度。

Var(X) = E[(X - μ)²] , where μ = E(X)

In words, you take each possible value, subtract the mean, square the result, multiply by its probability, and sum over all values.

换句话说，取每个可能取值，减去均值，平方，乘以对应概率，然后将所有结果相加。

A larger variance indicates that the values of X are more scattered around μ; a smaller variance means they are tightly clustered around the mean.

方差越大，表示X的取值在μ周围越分散；方差越小，表示取值紧密聚集在均值附近。

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4. The Shortcut Formula for Variance | 方差的简便计算公式

While the definition is important, the most efficient method for calculating variance in exams is the shortcut formula derived from expanding E[(X-μ)²].

虽然定义很重要，但在考试中最有效的计算方差的方法是扩展E[(X-μ)²]导出的简便公式。

Var(X) = E(X²) - [E(X)]²

Σ x² P(X=x). This formula avoids having to subtract the mean from each term individually.

其中，E(X²)是变量平方的期望值，计算公式为Σ x² P(X=x)。这个公式避免了逐一减去均值的麻烦。

then multiplying by probabilities, and finally summing. Do not square the sum of x P(X=x).

正确计算E(X²)至关重要：先对x值平方，再乘以概率，最后求和。切勿将x P(X=x)的和再平方。

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5. Properties of Variance: Var(aX + b) | 方差的性质：Var(aX + b)

Variance has important linear transformation properties. For constants a and b, the variance of a linear function of X is given by:

方差具有重要的线性变换性质。对于常数a和b，X的线性函数的方差由下式给出：

Var(aX + b) = a² Var(X)

Adding a constant b simply shifts the distribution without changing its spread, so b has no effect on the variance. Multiplying X by a scales the variance by a².

加上常数b只是平移分布而不改变分散程度，因此b对方差没有影响。将X乘以a会使方差乘以a²。

e negative sign inside the bracket does not change the result because squaring removes it.

例如，若Var(X)=3，则Var(2X - 5)=2²×3=12。注意括号内的负号不影响结果，因为平方会消去它。

d in combination with expected value: E(aX + b) = aE(X) + b. Do not confuse the two rules.

这个性质常与期望值结合考查：E(aX + b)=aE(X)+b。不要混淆这两个法则。

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6. Step-by-Step Calculation Using a Probability Distribution Table | 使用概率分布表逐步计算方差

Consider a discrete random variable X with the following probability distribution:

考虑一个离散随机变量X，其概率分布如下：

x	1	2	3	4
P(X=x)	0.1	0.3	0.4	0.2

To find Var(X), we first compute E(X) and E(X²).

为求Var(X)，我们先计算E(X)和E(X²)。

E(X) = Σ x P = 1×0.1 + 2×0.3 + 3×0.4 + 4×0.2 = 0.1 + 0.6 + 1.2 + 0.8 = 2.7

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

接下来，先将每个x值平方，再计算E(X²)：

E(X²) = Σ x² P = 1²×0.1 + 2²×0.3 + 3²×0.4 + 4²×0.2 = 0.1 + 1.2 + 3.6 + 3.2 = 8.1

Now apply the shortcut formula:

现在应用简便公式：

Var(X) = E(X²) - [E(X)]² = 8.1 - (2.7)² = 8.1 - 7.29 = 0.81

metic errors if you construct an extended table with columns for x, P(X=x), x·P, and x²·P.

这种方法系统性强，如果构建一个包含x、P(X=x)、x·P和x²·P的扩展表格，可以最大限度地减少计算错误。

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7. Variance and Standard Deviation | 方差与标准差

The standard deviation, denoted by σ or SD(X), is the positive square root of the variance. It is often more interpretable because it has the same units as the random variable.

标准差，记作σ或SD(X)，是方差的正平方根。它通常更容易解释，因为与随机变量具有相同的单位。

σ = √Var(X)

For the example above, the standard deviation is √0 This tells us that, on average, the values of X deviate from the mean of 2.7 by about 0.9.

对于上面的例子，标准差为√0.81=0.9。这告诉我们，X的取值平均偏离均值2.7约0.9个单位。

In Edexcel questions, you may be asked to find both the variance and the standard deviation, so always be prepared to take the square root.

在Edexcel试题中，可能要求同时求方差和标准差，因此要随时准备开平方根。

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8. Variance of a Discrete Uniform Distribution | 离散均匀分布的方差

A discrete uniform distribution occurs when all n possible values are equally likely. For a random variable X taking integer values from 1 to n, the variance has a concise formula.

当所有n个可能取值等可能时，就得到离散均匀分布。对于取值从1到n的整数的随机变量X，方差有一个简洁公式。

Var(X) = (n² - 1) / 12

the score on a fair six-sided die, then n = 6, and Var(X) = (36 - 1)/12 = 35/12 ≈ 2.9167.

例如，若X表示公平六面骰子的点数，则n=6，Var(X)=(36-1)/12=35/12≈2.9167。

This result can be derived using the shortcut formula and the sums of the first n integers and squares. It is useful to remember or be able to derive quickly.

该结果可利用简便公式以及前n个整数及其平方的求和推导出来。记住或能快速推导对解题有帮助。

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9. Sums and Differences of Independent Random Variables | 独立随机变量的和与差的方差

If X and Y are independent discrete random variables, the variance of their sum or difference follows a simple rule, regardless of the sign.

如果X和Y是独立的离散随机变量，其和或差的方差遵循一个简单规则，与符号无关。

Var(X + Y) = Var(X) + Var(Y)

Var(X - Y) = Var(X) + Var(Y)

he difference is the sum, not the difference, of the variances. This is because (-1)² = 1.

注意，差的方差是方差之和，而不是方差之差，这是因为(-1)²=1。

inear combinations: for independent variables X and Y, Var(aX + bY) = a²Var(X) + b²Var(Y).

这一性质推广到线性组合：对于独立变量X和Y，Var(aX + bY)=a²Var(X)+b²Var(Y)。

In Edexcel S1, you often use this rule when dealing with total scores from two independent experiments.

在Edexcel S1中，处理来自两个独立实验的总分时，常常要用到这个法则。

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10. Common Mistakes to Avoid | 常见错误及避免方法

Students often confuse E(X²) with [E(X)]². Remember: E(X²) is the average of the squared data, while [E(X)]² is the square of the average.

学生常常混淆E(X²)和[E(X)]²。记住：E(X²)是数据平方的平均值，而[E(X)]²是平均值的平方。

Another common error is forgetting to square the coefficient a in Var(aX + b). The correct factor is a², not a.

另一个常见错误是在Var(aX + b)中忘记平方系数a。正确的因子是a²，而不是a。

ectly assume Var(X - Y) = Var(X) - Var(Y). Always add variances for independent variables.

此外，一些考生错误地认为Var(X - Y)=Var(X)-Var(Y)。对于独立变量，方差始终相加。

Finally, when using the shortcut formula, ensure you are summing x²P, not (xP)². Extending the distribution table can prevent this slip.

最后，在使用简便公式时，确保求和的是x²P，而不是(xP)²。扩展分布表可以防止这种疏忽。

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11. Exam Tips for Edexcel | Edexcel考试技巧

(X)=Σ xP and Var(X)=E(X²)-[E(X)]². This allows method marks even if arithmetic goes wrong.

始终写下你使用的公式，例如E(X)=Σ xP和Var(X)=E(X²)-[E(X)]²。这样即使计算出错也能得到方法分。

Construct a clear table with columns x, P, xP, x²P when calculating variance. Show subtotals to minimise mistakes.

计算方差时，构建一个包含x、P、xP、x²P列的清晰表格。展示小计以减少错误。

Check that probabilities sum exactly to 1; if not, re-read the question. Also verify that your variance is non-negative – a negative variance signals an error.

检查概率之和是否为1；如果不是，重新审题。同样，验证方差是否为非负——负方差表明有错误。

If the question asks for standard deviation, present it to three significant figures unless otherwise instructed.

如果题目要求标准差，除非另有说明，结果保留三位有效数字。

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12. Summary | 总结

ared deviation from the mean and can be efficiently calculated using Var(X)=E(X²)-[E(X)]².

方差是衡量离散随机变量离散程度的基本度量。它定义为与均值的偏差平方的期望值，可以高效地使用Var(X)=E(X²)-[E(X)]²进行计算。

s, Var(X ± Y)=Var(X)+Var(Y). Standard deviation is simply the square root of the variance.

关键性质包括Var(aX+b)=a²Var(X)，以及对于独立变量，Var(X ± Y)=Var(X)+Var(Y)。标准差即方差的平方根。

By mastering these concepts and practicing with probability tables, you will be well-prepared for any Edexcel A-Level question on the variance of a discrete random variable.

掌握这些概念并通过概率表格进行练习，将让你为任何Edexcel A-Level离散随机变量方差的问题做好充分准备。

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