Exercise 19D.2: Expectation and Variance of a Discrete Random Variable | 练习19D.2：离散随机变量的期望与方差

📚 Exercise 19D.2: Expectation and Variance of a Discrete Random Variable | 练习19D.2：离散随机变量的期望与方差

In IB Mathematics, understanding the behaviour of a discrete random variable goes far beyond simply listing its possible values and probabilities. Exercise 19D.2 invites students to compute and interpret two fundamental measures: the expected value E(X), which represents the long‑run average of the variable, and the variance Var(X), which quantifies the spread of the distribution. This article walks through the theory, a detailed worked solution, graphical calculator tips, and common pitfalls, all tailored to the style of tasks found in this exercise.

在 IB 数学中，理解一个离散随机变量的行为远不止罗列出它的可能取值和对应概率。练习 19D.2 要求学生计算并解释两个基本度量：期望值 E(X)——代表变量的长期平均值，以及方差 Var(X)——量化分布的离散程度。本文梳理了相关理论、详细的解题过程、图形计算器技巧和常见错误，完全贴合该练习中所出现题目的风格。

1. Discrete Random Variables in Context | 离散随机变量的背景

A discrete random variable takes a countable number of values, each with a specific probability. The probability distribution must satisfy two conditions: every probability is between 0 and 1 inclusive, and the sum of all probabilities is exactly 1. Exercise 19D.2 typically presents such a distribution in tabular form and asks for E(X) and Var(X).

离散随机变量取可数个值，每个值对应一个特定的概率。概率分布必须满足两个条件：每个概率都在 0 到 1 之间（含端点），且所有概率之和恰好为 1。练习 19D.2 通常会以表格形式给出这样一个分布，并要求求出 E(X) 和 Var(X)。

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2. Setting Up the Problem from Exercise 19D.2 | 练习 19D.2 的问题设定

Let us work with a representative problem closely modelled on the exercises in this section. The random variable Y has the following distribution:

我们从一个与本部分习题高度相似的代表性问题入手。随机变量 Y 的分布如下：

y	0	1	2	3	4
P(Y = y)	0.1	0.2	0.2	0.4	0.1

The task is to calculate E(Y) and Var(Y), interpret the results, and explore how a linear transformation affects these quantities.

任务是要计算 E(Y) 与 Var(Y)，解释结果，并探讨线性变换如何影响这些量。

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3. Understanding Expected Value E(X) | 理解期望值 E(X)

The expected value of a discrete random variable is the probability‑weighted average of all its possible outcomes. It is not necessarily a value the variable can actually take; rather, it describes the centre of the distribution in a weighted sense.

离散随机变量的期望值是其所有可能结果按概率加权的平均值。它不一定是变量实际能取到的值，而是从加权角度描述了分布的中心位置。

Formula: E(X) = Σ [x · P(X = x)]

公式：E(X) = Σ [x · P(X = x)]

For the given distribution, we multiply each y value by its probability and sum the products. This calculation is straightforward but must be performed carefully to avoid arithmetic slips.

对于给定的分布，我们将每个 y 值乘以它的概率，再将乘积相加。这一计算虽然直接，但需要小心操作以避免算术错误。

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4. Step‑by‑Step Calculation of E(Y) | 逐步计算 E(Y)

We construct a working table that adds a row for the product y · P(Y = y):

我们构建一个计算表，增加一行 y · P(Y = y) 的乘积项：

y	0	1	2	3	4
P(Y=y)	0.1	0.2	0.2	0.4	0.1
y·P(Y=y)	0×0.1 = 0	1×0.2 = 0.2	2×0.2 = 0.4	3×0.4 = 1.2	4×0.1 = 0.4

Summing the product row gives: E(Y) = 0 + 0.2 + 0.4 + 1.2 + 0.4 = 2.2

将乘积行相加得到：E(Y) = 0 + 0.2 + 0.4 + 1.2 + 0.4 = 2.2

E(Y) = 2.2

Thus, if the experiment that generates Y were repeated a large number of times, the long‑run average outcome would be 2.2.

因此，如果产生 Y 的实验重复大量次数，其长期平均结果将为 2.2。

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5. Introducing Variance Var(X) | 引入方差 Var(X)

While the expected value tells us where the distribution is centred, the variance measures how far the values typically spread from that centre. A larger variance indicates greater dispersion; a variance of zero would mean the variable is constant.

期望值告诉我们分布中心在哪里，而方差度量的是数值通常离该中心有多远。方差越大，离散程度越高；方差为零则意味着变量是常数。

The definitional formula is Var(X) = E[(X – nal form is often faster and less prone to rounding errors, especially in IB examinations.

定义公式为 Var(X) = E[(X – μ)²]，其中 μ = E(X)。然而，等价的简便计算形式通常更快且不易引入舍入误差，尤其在 IB 考试中。

Var(X) = E(X²) – [E(X)]²

This is the version emphasised in Exercise 19D.2 and is derived from expanding the squared deviation and applying linearity of expectation.

这就是练习 19D.2 强调的版本，它通过对偏差平方展开并应用期望的线性性质推导出来。

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6. Computing E(Y²) and Var(Y) | 计算 E(Y²) 与 Var(Y)

), the expected value of the squared variable. We extend the table to include y²·P(Y = y):

首先我们计算 E(Y²)，即平方变量的期望值。扩展表格以包含 y²·P(Y = y)：

y	0	1	2	3	4
P(Y=y)	0.1	0.2	0.2	0.4	0.1
y²	0	1	4	9	16
y²·P(Y=y)	0×0.1 = 0	1×0.2 = 0.2	4×0.2 = 0.8	9×0.4 = 3.6	16×0.1 = 1.6

Adding the last row: E(Y²) = 0 + 0.2 + 0.8 + 3.6 + 1.6 = 6.2

将最后一行相加：E(Y²) = 0 + 0.2 + 0.8 + 3.6 + 1.6 = 6.2

Now apply the short formula: Var(Y) = E(Y²) – [E(Y)]² = 6.2 – (2.2)² = 6.2 – 4.84 = 1.36

现在应用简捷公式：Var(Y) = E(Y²) – [E(Y)]² = 6.2 – (2.2)² = 6.2 – 4.84 = 1.36

Var(Y) = 1.36

The standard deviation is σ = √1 ee decimal places), offering a more interpretable measure of spread in the original units.

标准差为 σ = √1.36 ≈ 1.166（保留三位小数），在原始单位下给出了更易于解释的离散程度。

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7. Interpreting the Results in Context | 在语境中解释结果

ts expectation near the middle of the range. Here the expected value 2.2 is slightly right‑ typical outcomes will often be between about 1.0 and 3.4 (mean ± one standard deviation).

一个公平游戏或对称分布通常期望值接近取值范围的中部。此时期望值 2.2 相较于中位数（2）略微右移，反映出了 y=3 处概率较大。方差 1.36 表明，与均值偏差的平方平均为 1.36，这意味着典型结果通常落在均值 ± 一个标准差的范围内，即约在 1.0 到 3.4 之间。

Understanding this variation is crucial in contexts like insurance premiums or quality control, where the expected outcome must be balanced against the risk (spread) involved.

在诸如保险费或质量控制的场景中，理解这种波动至关重要，因为期望结果必须与所涉及的风险（离散程度）进行权衡。

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8. Using a Graphical Display Calculator (GDC) | 使用图形计算器 (GDC)

IB candidates are encouraged to verify their hand calculations with a GDC. For a TI‑Nspire or TI‑84, the procedure is to enter the values in one list (e.g. L1) and the corresponding probabilities in another list (L2), then use the 1‑Var Stats command with frequency list L2. The output directly provides Σx (sum), mean x̄ (which is E(X)), Σx², and σx (population standard deviation, from which variance is obtained by squaring).

IB 考生被鼓励使用图形计算器来检驗他们的手算结果。对于 TI‑Nspire 或 TI‑84，操作步骤是将数值输入一个列表（如 L1），将相应概率输入另一个列表（L2），然后使用 1‑Var Stats 命令并指定频率列表为 L2。输出结果会直接给出 Σx（和）、均值 x̄（即 E(X)）、Σx² 以及 σx（总体标准差，平方后即得方差）。

When using a calculator, be careful to select the population standard deviation σ, not the sample standard deviation Sx, because you are working with the complete theoretical probability distribution and not a sample.

使用计算器时要小心，应选择总体标准差 σ，而非样本标准差 Sx，因为你处理的是完整的理论概率分布，而非样本。

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9. Linear Transformations E(aX + b) and Var(aX + b) | 线性变换的期望与方差

Exercise 19D.2 frequently extends the basic calculation to linear combinations of random variables. The rules are essential: for constants a and b,

练习 19D.2 经常将基本计算延伸到随机变量的线性组合。下列规则至关重要：对于常数 a 与 b，

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

Var(aX + b) = a² Var(X)

The expectation is linear, so adding a constant shifts the mean by that constant; multiplying by a scales both the mean and the spread. The variance, however, is not affected by the additive constant b because spread is unchanged by a simple translation. The multiplicative factor a enters as a², reflecting that variance involves squared deviations.

期望是线性的，因此加上常数会使均值平移该常数量；乘以因子 a 会同时缩放均值和离散范围。然而方差不受加法常数 b 的影响，因为单纯的平移不会改变离散程度。乘法因子 a 以 a² 的形式进入，体现出方差涉及偏差平方的性质。

As an illustration: if W = 3Y – 1, then E(W) = 3 × 2.2 – 1 = 5.6, and Var(W) = 3² × 1.36 = 12.24.

举例说明：若 W = 3Y – 1，则 E(W) = 3 × 2.2 – 1 = 5.6，Var(W) = 3² × 1.36 = 12.24。

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10. Common Mistakes When Tackling Exercise 19D.2 | 解决练习 19D.2 时的常见错误

One frequent error is forgetting to square the constant when computing Var(aX + b). Students often write a · Var(X) instead of a² · Var(X). Another is mis‑applying the computational formula by subtracting E(X²) from [E(X)]², which gives a negative variance—an immediate warning of a mistake.

一个常见错误是在计算 Var(aX + b) 时忘记将常数平方。学生经常写成 a · Var(X) 而不是 a² · Var(X)。另一个错误是误用简便计算公式，用 E(X²) 减去 [E(X)]² 时颠倒了顺序，导致出现负的方差——这是错误的明确信号。

Also, when the probability distribution is given in a histogram or as a function, students may misinterpret the values on the x‑axis. Always ensure the sum of probabilities adds to exactly 1; if a table is incomplete, use the complement rule to find the missing probability before calculating E(X) and Var(X).

此外，当概率分布以直方图或函数形式给出时，学生可能误读 x 轴上的数值。务必确保概率之和恰好为 1；如果表格不完整，在计算 E(X) 和 Var(X) 之前应使用补集法则求出缺失的概率。

Finally, remember that the mean and variance describe the distribution; they do not give information about individual outcomes unless combined with a known probability model like the binomial or normal distribution.

最后，记住均值和方差描述的是分布本身；除非与已知的概率模型（如二项分布或正态分布）结合，否则它们不提供关于个别结果的信息。

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11. Summary and Links to IB Assessment | 总结及与 IB 评估的联系

Exercise 19D.2 trains students to move comfortably between tabular probability data, algebraic expectations, and variance calculations. The skills assessed here feed directly into Paper 1 and Paper 2 questions on discrete distributions, and they lay the groundwork for later topics such as the binomial distribution and hypothesis testing.

练习 19D.2 训练学生熟练地在表格概率数据、代数期望以及方差计算之间切换。这里考察的技能直接应用于试卷一和试卷二中有关离散分布的问题，并为后期主题如二项分布和假设检验奠定基础。

By mastering the core procedures—setting up a clear table, applying E(X) = Σ x·p, using Var(X) = E(X²) – [E(X)]², and understanding how linear transformations affect expectation and variance—students will approach related examination tasks with confidence.

通过掌握核心流程——构建清晰的表格、应用 E(X) = Σ x·p、使用 Var(X) = E(X²) – [E(X)]²，并理解线性变换如何影响期望与方差——学生将能够自信地应对相关考试任务。

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