Discrete Random Variables: A Comprehensive IB Maths Guide | 离散随机变量：IB数学全面指南

📚 Discrete Random Variables: A Comprehensive IB Maths Guide | 离散随机变量：IB数学全面指南

In IB Mathematics, whether you follow the Analysis and Approaches (AA) or Applications and Interpretation (AI) syllabus, discrete random variables form the foundation of probability and statistics. Understanding how to model random phenomena with discrete probability distributions, how to calculate expected values and variances, and how to work with common families such as the binomial and Poisson distributions is essential for exam success.

在IB数学中，无论你学习的是分析与方法(AA)还是应用与解释(AI)课程，离散随机变量都是概率与统计的基础。掌握如何用离散概率分布对随机现象建模、如何计算期望值和方差，以及如何运用二项分布、泊松分布等常见分布族，对于考试成功至关重要。

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1. What is a Random Variable? | 什么是随机变量？

A random variable is the numerical outcome of a random phenomenon. Instead of describing events in words, we assign a number to each possible outcome. For example, when tossing a fair coin three times, the number of heads obtained is a random variable. Formally, a random variable X is a function from the sample space to the real numbers.

随机变量是随机现象的数字结果。我们不是用文字描述事件，而是给每个可能的结果分配一个数字。例如，抛一枚公平硬币三次，得到的正面次数就是一个随机变量。形式上，随机变量 X 是从样本空间到实数的函数。

lar value it can take by the corresponding lower-case letter, such as x. So P(X = x) means 'the probability that the random variable X takes the value x'. This notation is used throughout IB probability questions.

我们通常用大写字母如 X、Y 或 Z 表示随机变量，用对应的小写字母如 x 表示它可能取的某个具体值。因此 P(X = x) 表示"随机变量 X 取值 x 的概率"。这一记号贯穿整个IB概率题目。

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2. Discrete vs Continuous Random Variables | 离散与连续随机变量

Random variables can be discrete or continuous. A discrete random variable takes a countable number of distinct values, such as 0, 1, 2, 3, ... . These values usually come from counting: the number of goals in a match, the number of defective items in a box, or the score on a single die. A continuous random variable, by contrast, can take any value within an interval, such as the height of a student or the time taken to run 100 m.

随机变量可以是离散的或连续的。离散随机变量取可数个不同的值，如 0, 1, 2, 3, ...。这些值通常来自计数：一场比赛中的进球数、一箱产品中的次品数，或掷一颗骰子得到的点数。相反，连续随机变量可以在一个区间内取任何值，例如学生的身高或跑100米所用的时间。

Feature	Discrete	Continuous
Values	Countable, often integers	Any value in an interval
Probability tool	Probability mass function	Probability density function
P(X = x)	Usually greater than 0	Equal to 0 for every single x

In this article we focus on discrete random variables, which are the ones examined most directly in IB Paper 1 and Paper 2 questions on probability distributions.

本文我们专注于离散随机变量，它也是IB Paper 1 和 Paper 2 概率分布题目中最常直接考查的对象。

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3. Probability Distributions and the Probability Mass Function | 概率分布与概率质量函数

t X takes that value. We write the probability that X equals x as P(X = x). This can be presented in a table, in a formula, or in a graph.

离散随机变量 X 由其概率分布来描述，该分布列出 X 的每个可能取值以及 X 取该值的概率。我们将 X 等于 x 的概率记作 P(X = x)。这可以用表格、公式或图形来表示。

A valid discrete probability distribution must satisfy two conditions: every probability must lie between 0 and 1 inclusive, and the sum of all probabilities must equal 1. These two facts are often used in exam questions to find an unknown constant.

一个合法的离散概率分布必须满足两个条件：每个概率都必须介于0与1之间（含端点），且所有概率之和必须等于1。这两个条件常常被用于考试题中求未知常数。

0 ≤ P(X = x) ≤ 1 and Σ P(X = x) = 1

or x = 1, 2, 3, for example, you can find k by requiring that k(1) + k(2) + k(3) = 1, so 6k = 1 and k = 1/6.

给出每个 x 对应 P(X = x) 的函数称为概率质量函数（pmf），常写作 p(x) 或 f(x)。例如，若已知 p(x) = kx，x = 1, 2, 3，则由 k(1) + k(2) + k(3) = 1 可得 6k = 1，所以 k = 1/6。

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4. Worked Example: Three Fair Coins | 实例：抛三枚公平硬币

. The probabilities are P(X = 0) = 1/8, P(X = 1) = 3/8, P(X = 2) = 3/8 and P(X = 3) = 1/8.

考虑随机变量 X = 抛三枚公平硬币所得到的正面数。可能的取值为 0, 1, 2, 3。概率为 P(X = 0) = 1/8, P(X = 1) = 3/8, P(X = 2) = 3/8, P(X = 3) = 1/8。

This distribution can be summarised in a table, which is the standard way to present a discrete probability distribution in IB exams:

这个分布可以总结成一个表格，这也是IB考试中呈现离散概率分布的标准方式：

x	0	1	2	3
P(X = x)	1/8	3/8	3/8	1/8

ch as P(X ≥ 2) = 3/8 + 1/8 = 1/2, or P(1 ≤ X ≤ 2) = 3/8 + 3/8 = 3/4. Notice how the phrase 'at least two' translates into the inequality X ≥ 2.

概率之和为1，符合要求。从表中我们可以读出诸如 P(X ≥ 2) = 3/8 + 1/8 = 1/2，或 P(1 ≤ X ≤ 2) = 3/8 + 3/8 = 3/4 等事件的概率。注意"至少两个"这样的表述如何转化为不等式 X ≥ 2。

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5. The Cumulative Distribution Function | 累积分布函数

p to and including the value x. For the three-coin example, F(0) = 1/8, F(1) = 4/8 = 1/2, F(2) = 7/8 and F(3) = 1.

离散随机变量的累积分布函数（cdf）定义为 F(x) = P(X ≤ x)，它给出累积到并包含取值 x 的总概率。对于抛三枚硬币的例子，F(0) = 1/8, F(1) = 4/8 = 1/2, F(2) = 7/8, F(3) = 1。

). It is always non-decreasing, tends to 0 as x becomes very small and tends to 1 as x becomes very large.

离散随机变量的累积分布函数是一个阶梯函数：在相邻可能取值之间保持常数，并在每个取值处跳跃，跳跃高度等于 P(X = x)。它总是单调不减的，当 x 取很小时趋近于0，当 x 取很大时趋近于1。

F(2) − F(0) = 7/8 − 1/8 = 3/4, which matches the answer obtained directly from the table.

用累积分布函数求 P(a ≤ X ≤ b) 时，使用 F(b) − F(a − 1)。例如 P(1 ≤ X ≤ 2) = F(2) − F(0) = 7/8 − 1/8 = 3/4，与直接从表格得到的结果一致。

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6. Expected Value (Mean) | 期望值（均值）

The expected value of a discrete random variable is its long-run average value. It is calculated by multiplying each possible value by its probability and adding the results. The expected value is denoted by E(X) or by the Greek letter μ (mu).

离散随机变量的期望值是它的长期平均值。计算方法是将每个可能取值乘以对应概率，再把结果相加。期望值记作 E(X) 或用希腊字母 μ（mu）表示。

E(X) = Σ x · P(X = x)

Notice that 1.5 is not a value that X can actually take; the expected value is an average, not an actual outcome.

对于抛三枚硬币的例子，E(X) = 0 × 1/8 + 1 × 3/8 + 2 × 3/8 + 3 × 1/8 = 12/8 = 1.5。注意 1.5 并不是 X 实际能取的值；期望值是平均值，而不是某次实际结果。

Expected value is useful when comparing options, such as choosing between two games or two insurance policies. The option with the higher expected value is better in the long run, provided the risk (variance) is acceptable.

在比较不同方案（例如两个游戏或两种保险方案）时，期望值很有用。长期来看，期望值更高的方案更优，前提是其风险（方差）可以接受。

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

The variance measures how spread out the values of a random variable are around the mean. It is defined as the expected value of the squared deviation from the mean, and it is denoted by Var(X) or by σ² (sigma squared).

方差衡量随机变量取值围绕均值的分散程度。它定义为取值与均值偏差平方的期望，记作 Var(X) 或 σ²（sigma 平方）。

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Var(X) = E(X²) − [E(X)]²

oin example, E(X²) = 0 × 1/8 + 1 × 3/8 + 4 × 3/8 + 9 × 1/8 = 24/8 = 3, so Var(X) = 3 − 1.5² = 3 − 2.25 = 0.75.

这个计算式通常比定义式更快。先求 E(X²) = Σ x² · P(X = x)，再减去 E(X) 的平方。对于抛三枚硬币的例子，E(X²) = 0 × 1/8 + 1 × 3/8 + 4 × 3/8 + 9 × 1/8 = 24/8 = 3，所以 Var(X) = 3 − 1.5² = 3 − 2.25 = 0.75。

The standard deviation is the square root of the variance: σ = √Var(X). Here σ = √0 tion is measured in the same units as X, which makes it easier to interpret than variance.

标准差是方差的平方根：σ = √Var(X)。这里 σ = √0.75 ≈ 0.866。标准差与 X 的单位相同，因此比方差更便于解释。

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8. Linear Transformations of a Discrete Random Variable | 离散随机变量的线性变换

aX + b, where a and b are constants, then the mean and variance transform in a simple way.

考试题常问：当随机变量被变换后，均值和方差会怎样变化。若 Y = aX + b，其中 a 与 b 为常数，则均值和方差有简单的变换规律。

E(aX + b) = aE(X) + b and Var(aX + b) = a² Var(X)

Adding a constant b shifts every value by b, so the mean shifts by b but the spread does not change. Multiplying by a scales every value, so the mean is multiplied by a and the variance is multiplied by a². The standard deviation is multiplied by |a|, not by a.

加上常数 b 会使每个取值平移 b，因此均值平移 b，但分散程度不变。乘以 a 会把每个取值按比例缩放，因此均值乘以 a，方差乘以 a²。标准差乘以 |a|，而不是 a。

2X − 5, then E(Y) = 2(4) − 5 = 3 and Var(Y) = 2² × 3 = 12. The standard deviation of Y is √12 ≈ 3.46, not 2 × 1.73 = 3.46 — the same value, but reached by taking the absolute value of a.

例如，若 X 满足 E(X) = 4，Var(X) = 3，且 Y = 2X − 5，则 E(Y) = 2(4) − 5 = 3，Var(Y) = 2² × 3 = 12。Y 的标准差是 √12 ≈ 3.46，而不是 2 × 1.73 = 3.46 ——数值相同，但要通过对 a 取绝对值得到。

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9. The Binomial Distribution | 二项分布

The binomial distribution models the number of successes in n independent trials, where each trial has only two outcomes (success or failure) and the probability of success p is constant. We write X ~ B(n, p).

二项分布用于建模 n 次独立试验中的成功次数，其中每次试验只有两种结果（成功或失败），且成功概率 p 保持不变。记作 X ~ B(n, p)。

P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ, E(X) = np, Var(X) = np(1 − p)

For example, if X ~ you can check these using the binomial probability functions on your graphic display calculator.

例如，若 X ~ B(10, 0.3)，则 P(X = 3) = C(10, 3) × 0.3³ × 0.7⁷ ≈ 0.2668，E(X) = 10 × 0.3 = 3，Var(X) = 10 × 0.3 × 0.7 = 2.1。考试中你可以用图形计算器上的二项概率函数来检验这些结果。

Always check the four binomial conditions before using this model: a fixed number of trials n, two possible outcomes per trial, a constant probability p, and independence between trials.

在使用该模型前，务必检查四个二项条件：试验次数 n 固定、每次试验只有两种结果、概率 p 保持不变、各次试验相互独立。

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10. The Poisson Distribution | 泊松分布

The Poisson distribution models the number of occurrences of an event in a fixed interval of time, length, area or volume, when the events occur independently and at a constant average rate λ (lambda). We write X ~ Po(λ).

泊松分布用于建模在固定时间、长度、面积或体积区间内某事件发生的次数，前提是事件相互独立且以恒定平均速率 λ（lambda）发生。记作 X ~ Po(λ)。

P(X = r) = e⁻λ λʳ ÷ r!, E(X) = λ, Var(X) = λ

For example, if X ~ Po(4), then P(X = 2) = e⁻⁴ × 4² ÷ 2! ture of the Poisson distribution is that its mean and variance are equal, both equal to λ.

例如，若 X ~ Po(4)，则 P(X = 2) = e⁻⁴ × 4² ÷ 2! = 8e⁻⁴ ≈ 0.1465，且 P(X ≥ 1) = 1 − P(X = 0) = 1 − e⁻⁴ ≈ 0.9817。泊松分布的一个有用性质是它的均值与方差相等，都等于 λ。

rses. The Poisson distribution can also be used to approximate the binomial distribution when n is large and p is small, with λ = np.

泊松分布主要在IB应用与解释(AI) HL中学习，而二项分布则同时出现在AA与AI课程中。当 n 很大且 p 很小时，泊松分布也可用来近似二项分布，此时 λ = np。

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11. Choosing the Right Model and Exam Technique | 如何选择模型与考试技巧

Many marks are lost not because of calculation errors but because the wrong distribution is chosen. Ask yourself whether the situation involves a fixed number of trials with two outcomes (binomial) or a count of events in a continuous interval with a given rate (Poisson).

很多失分并非因为计算错误，而是因为选错了分布。要问自己：题目涉及的是固定次数的两种结果试验（二项），还是在连续区间内以给定速率发生的事件计数（泊松）。

Question clue	Model
'out of 20 students', 'in 15 tosses'	Binomial B(n, p)
'per hour', 'per page', 'per km'	Poisson Po(λ)
Table of x and P(X = x) given	General discrete distribution

When answering, always define the random variable in words, state the distribution and its parameters, and show the probability expression before evaluating it. This earns method marks even if the final value is wrong. Use your calculator's binomial and Poisson functions to check answers efficiently.

作答时，务必先用文字定义随机变量，写出分布及其参数，并展示概率表达式再进行计算。即使最终数值有误，这也能拿到方法分。用计算器的二项与泊松函数可以高效检验答案。

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12. Common Mistakes and Summary | 常见错误与总结

E(X²) − [E(X)]² with E(X²) = [E(X)]², or taking the square root when variance is required.

一个常见错误是：当分布中含有未知常数时，忘记所有概率之和必须等于1。另一个错误是把 Var(X) = E(X²) − [E(X)]² 与 E(X²) = [E(X)]² 弄混，或在题目要求方差时错误地取了平方根。

For binomial and Poisson questions, check the conditions carefully. For Poisson, remember that λ is the mean for the interval stated; if the interval changes, λ must be scaled proportionally. For binomial, remember that mean and variance are not equal in general.

对于二项与泊松题目，要仔细检查条件。对于泊松分布，记住 λ 是题目所述区间内的均值；若区间改变，λ 必须按比例缩放。对于二项分布，记住均值与方差一般不相等。

) = E(X²) − [E(X)]²; linear transformations follow E(aX + b) = aE(X) + b and Var(aX + b) = a² Var(X); and the binomial and Poisson distributions provide powerful models for counts in IB examinations.

总结：离散随机变量取可数的值，其概率之和为1；均值为 E(X) = Σ x P(X = x)；方差为 Var(X) = E(X²) − [E(X)]²；线性变换满足 E(aX + b) = aE(X) + b 与 Var(aX + b) = a² Var(X)；二项分布与泊松分布为IB考试中的计数问题提供了强有力的模型。

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