📚 Exercise 18A: Discrete Random Variables | 练习18A：离散随机变量
In IB Mathematics, mastering discrete random variables is essential for understanding probability distributions and statistical inference. Exercise 18A focuses on the foundational concepts: constructing probability mass functions, calculating expected values, and determining variances and standard deviations. This article provides a comprehensive revision guide covering all key topics you will encounter in this exercise, with step-by-step examples and exam tips.
在 IB 数学中，掌握离散随机变量对于理解概率分布和统计推断至关重要。练习 18A 聚焦于基本概念：构建概率质量函数、计算期望值以及确定方差和标准差。本文提供一份全面的复习指南，涵盖该练习中你将遇到的所有关键主题，并配有逐步解析的例题和考试技巧。
1. Introduction to Discrete Random Variables | 离散随机变量简介
A discrete random variable is a variable whose value depends on the outcome of a random event and can only take a countable number of distinct values. For example, the number of heads when tossing three coins or the sum of two dice are discrete random variables.
离散随机变量是一种其取值依赖于随机事件结果且只能取可数个不同值的变量。例如，抛掷三枚硬币时正面朝上的次数，或者两个骰子的点数之和，都是离散随机变量。
Unlike continuous random variables, which can take any value within an interval, discrete random variables have gaps between the possible values. This distinction is fundamental to choosing the correct analytical methods.
与可以在区间内取任意值的连续随机变量不同，离散随机变量的可能取值之间存在间隙。这一区别对于选择正确的分析方法至关重要。
2. Probability Mass Function (PMF) | 概率质量函数
The probability mass function (PMF) of a discrete random variable X gives the probability that X equals a specific value x. It is often written as P(X = x) = p(x), where p(x) is the probability associated with the outcome x.
离散随机变量 X 的概率质量函数 (PMF) 给出了 X 等于特定值 x 的概率。通常记作 P(X = x) = p(x)，其中 p(x) 是与结果 x 相关的概率。
To construct a PMF, list all possible values of X together with their corresponding probabilities. This is often presented in a table, making it easy to read and verify conditions.
要构建一个 PMF，需列出 X 的所有可能取值以及相应的概率。通常会以表格形式呈现，这样便于阅读和验证条件。
3. Conditions for a Valid PMF | 概率质量函数的有效条件
For a function to serve as a valid PMF, it must satisfy two critical conditions: (i) 0 ≤ P(X = x) ≤ 1 for every value x, and (ii) the sum of all probabilities equals 1, i.e., Σ P(X = x) = 1.
一个函数要成为有效的 PMF，必须满足两个关键条件：(i) 对于每个值 x，0 ≤ P(X = x) ≤ 1；(ii) 所有概率之和等于 1，即 Σ P(X = x) = 1。
If either condition is violated, the function does not represent a legitimate probability distribution. Always check these conditions when given or deriving a PMF.
如果任一条件被违反，该函数就不代表一个合法的概率分布。在给定或推导 PMF 时，务必检查这些条件。
4. Expected Value E(X) | 期望值 E(X)
The expected value (or mean) of a discrete random variable X is a measure of its central tendency. It is calculated by summing each possible value multiplied by its probability:
离散随机变量 X 的期望值（或均值）是其集中趋势的度量。它通过将每个可能取值乘以其概率然后求和来计算：
E(X) = Σ x · P(X = x)
The expected value is not necessarily one of the actual outcomes—it is a weighted average of all possibilities. In IB problems, you will often need to compute E(X) from a table and then use it to find variance.
期望值不一定是某个实际出现的结果——它是所有可能结果的加权平均值。在 IB 题目中，你经常需要从表格中计算 E(X)，然后利用它来求方差。
5. Variance and Standard Deviation | 方差与标准差
Variance measures the spread of the random variable around its expected value. The formula for variance Var(X) is:
方差衡量随机变量围绕其期望值的分散程度。方差 Var(X) 的公式为：
Var(X) = Σ (x – μ)² · P(X = x)
where μ = E(X). A more computationally convenient form is:
其中 μ = E(X)。另一个计算更方便的形式是：
Var(X) = E(X²) – [E(X)]²
To use this, first compute E(X²) = Σ x² · P(X = x), then subtract the square of the expected value. The standard deviation is the positive square root of the variance: σ = √Var(X). Both are frequently assessed in IB examinations.
使用该公式时，先计算 E(X²) = Σ x² · P(X = x)，然后减去期望值的平方。标准差是方差的正平方根：σ = √Var(X)。两者在 IB 考试中经常被评估。
6. Linear Transformations of a Random Variable | 随机变量的线性变换
If a new random variable Y is defined by a linear transformation of X, i.e., Y = aX + b, then the expected value and variance transform predictably:
如果一个新的随机变量 Y 由 X 的线性变换定义，即 Y = aX + b，那么期望值和方差会按可预测的方式变换：
E(Y) = a E(X) + b
Var(Y) = a² Var(X)
Notice that adding a constant b shifts the mean but does not affect the variance, while multiplying by a scales the mean by a and the variance by a². These rules are extremely useful when analysing transformed data or game scores.
注意，加上常数 b 会移动均值但不影响方差，而乘以 a 会将均值缩放 a 倍，将方差缩放 a² 倍。这些规则在分析变换后的数据或游戏得分时非常有用。
The standard deviation of Y is simply |a| times the standard deviation of X, because σᵧ = √Var(Y) = √(a²Var(X)) = |a| σₓ.
Y 的标准差就是 X 的标准差乘以 |a|，因为 σᵧ = √Var(Y) = √(a²Var(X)) = |a| σₓ。
7. Sums of Independent Random Variables | 独立随机变量之和
When working with two independent discrete random variables X and Y, the following properties hold for their sum:
当处理两个独立的离散随机变量 X 和 Y 时，它们的和具有以下性质：
E(X + Y) = E(X) + E(Y)
Var(X + Y) = Var(X) + Var(Y)
The expectation of the sum is the sum of expectations, regardless of independence. However, the variance sum property holds only when X and Y are independent. If they are not independent, the covariance must be considered, but Exercise 18A typically deals with independent cases.
和的期望值是期望值之和，无论是否独立。然而，方差的和性质仅在 X 和 Y 独立时成立。如果不独立，就必须考虑协方差，但练习 18A 通常处理独立的情形。
These rules extend to more than two variables and are particularly handy when finding the mean and variance of a total score from repeated independent trials.
这些规则可以推广到两个以上的变量，并且在求多次独立试验总分的均值和方差时特别好用。
8. Worked Example from Exercise 18A | 练习18A 中的例题
Problem: A fair four-sided die (with faces 1, 2, 3, 4) is rolled twice. Let X be the larger of the two numbers rolled. Find the probability distribution of X, and hence calculate E(X) and Var(X).
问题：抛掷一个公平的四面骰子（面为 1,2,3,4）两次。设 X 为两次抛掷中较大的数。求 X 的概率分布，并由此计算 E(X) 和 Var(X)。
First, we list all 16 equally likely outcomes of two rolls and determine the maximum value for each. By counting, we obtain the PMF shown in the table below.
首先，我们列出两次抛掷中所有 16 个等可能结果，并确定每次的最大值。通过计数，我们得到如下表所示的 PMF。
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| P(X=x) | 1/16 | 3/16 | 5/16 | 7/16 |
Verify: 1/16 + 3/16 + 5/16 + 7/16 = 16/16 = 1, so the PMF is valid.
验证：1/16 + 3/16 + 5/16 + 7/16 = 16/16 = 1，因此 PMF 有效。
Now calculate E(X):
现在计算 E(X)：
E(X) = 1·(1/16) + 2·(3/16) + 3·(5/16) + 4·(7/16) = (1 + 6 + 15 + 28)/16 = 50/16 = 3.125
Next, compute E(X²):
接下来，计算 E(X²)：
E(X²) = 1²·(1/16) + 4·(3/16) + 9·(5/16) + 16·(7/16) = (1 + 12 + 45 + 112)/16 = 170/16 = 10.625
Then variance using the short-cut formula:
然后使用简便公式求方差：
Var(X) = E(X²) – [E(X)]² = 10.625 – (3.125)² = 10.625 – 9.765625 = 0.859375
Thus, E(X) = 3.125 and Var(X) ≈ 0.859. The standard deviation is √0.859375 ≈ 0.927. These results align with typical Exercise 18A tasks.
因此，E(X) = 3.125，Var(X) ≈ 0.859。标准差为 √0.859375 ≈ 0.927。这些结果与练习 18A 的典型任务相符。
9. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Mistake 1: Forgetting to check that the total probability sums to 1. Always do a quick sanity check before calculating expected values.
错误一：忘记检查总概率之和是否等于 1。在计算期望值之前，务必做一个快速的合理性检查。
Mistake 2: Using the wrong formula for variance. Students often forget to square the values when computing E(X²) or incorrectly use E(X²) – E(X). Remember: Var(X) = E(X²) – [E(X)]².
错误二：使用错误的方差公式。学生在计算 E(X²) 时常常忘记将取值平方，或者错误地使用 E(X²) – E(X)。请记住：Var(X) = E(X²) – [E(X)]²。
Mistake 3: Misapplying linear transformation rules. Multiplying X by a changes the variance by a², not by a. Double-check your use of a and a².
错误三：错误运用线性变换规则。将 X 乘以 a，方差会变化 a² 倍，而不是 a 倍。请双重核对 a 和 a² 的使用。
Mistake 4: Assuming independence when combining variances. If the problem does not state independence, do not use Var(X + Y) = Var(X) + Var(Y). In Exercise 18A, independence is usually given, but always read carefully.
错误四：在合并方差时假设独立。如果题目未说明独立，切勿使用 Var(X + Y) = Var(X) + Var(Y)。在练习 18A 中，通常会给定独立性，但一定要仔细阅读题目。
10. Key Takeaways for IB Exams | IB 考试要点
Discrete random variables appear regularly in IB Mathematics paper 1 and paper 2 questions. The examiners often test your ability to construct probability distributions from a real-life scenario, compute E(X) and Var(X), and interpret these values in context.
离散随机变量经常出现在 IB 数学试卷一和试卷二的问题中。考官经常测试你从现实场景构建概率分布、计算 E(X) 和 Var(X) 并在上下文中解读这些数值的能力。
Be prepared for questions that combine probability with algebra, where part of the distribution is unknown, and you must use Σ P = 1 or E(X) to find missing probabilities.
要做好准备应对将概率与代数结合的问题，即部分分布未知，你必须利用 Σ P = 1 或 E(X) 来求解缺失的概率。
Finally, always present your solution clearly using tables for PMF and show all steps for variance calculations. This not only ensures marks for method but also reduces arithmetic errors.
最后，务必使用表格清晰呈现 PMF，并展示方差计算的所有步骤。这不仅能确保拿到方法分，还能减少算术错误。
11. Practice Tips for Mastery | 掌握练习技巧
To excel in Exercise 18A and similar topics, create your own random variable scenarios and practice deriving PMFs from scratch. Work with dice, spinners, or card draws to build intuition.
要在练习 18A 及类似主题中取得优异成绩，请自行构建随机变量场景，并练习从零开始推导 PMF。使用骰子、转盘或抽牌来建立直觉。
Regularly time yourself when solving full-length questions, as IB exams require both speed and accuracy. Use the relationships E(X²) and Var(X) to cross-check results.
在解答完整题目时定期计时，因为 IB 考试既要求速度也要求准确性。利用 E(X²) 与 Var(X) 之间的关系来交叉核对结果。
Review transformations and sum properties until you can apply them without hesitation. A solid grasp of these concepts will serve you well in later topics like the binomial distribution and continuous random variables.
不断复习变换与和的性质，直到能够毫不犹豫地应用它们。扎实掌握这些概念将有助于你后续学习二项分布和连续随机变量等主题。
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