IB Mathematics: Core Concepts of Discrete Random Variables | IB数学：离散随机变量核心概念

📚 IB Mathematics: Core Concepts of Discrete Random Variables | IB数学：离散随机变量核心概念

A discrete random variable is one of the most fundamental topics in IB Mathematics, appearing in both Analysis & Approaches (AA) and Applications & Interpretation (AI) courses. Understanding its core concepts — probability distributions, expected value, and variance — is essential for success in Paper 1, Paper 2, and the internal assessment. This article provides a systematic review of the key ideas, formulas, and exam-style applications.

离散随机变量是IB数学中最基础的话题之一，在分析与方法（AA）和应用与解释（AI）两门课程中均有涉及。理解其核心概念——概率分布、期望值和方差——对于Paper 1、Paper 2以及内部评估的成功都至关重要。本文系统梳理了关键思想、公式和考试风格的应用。

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

A random variable is a variable whose possible values are numerical outcomes of a random experiment. A discrete random variable takes a countable number of distinct values, typically integers. For example, the number of heads when tossing three coins, or the number of customers arriving at a shop in an hour.

随机变量是其可能取值为随机试验数值结果的变量。离散随机变量取可数个不同的值，通常是整数。例如，掷三枚硬币时出现正面的次数，或一小时内到达商店的顾客数。

We denote a discrete random variable by an uppercase letter, such as X, and its specific values by lowercase letters, such as x. The set of all possible values of X is called the support of X.

我们用大写字母如X表示离散随机变量，用小写字母如x表示其具体取值。X的所有可能值的集合称为X的支集。

Example: Let X be the sum of two fair dice. Possible values: 2, 3, 4, …, 12.
示例：设X为两颗公平骰子的点数之和。可能取值：2, 3, 4, …, 12。

2. Probability Distribution Function | 概率分布函数

The probability distribution of a discrete random variable is a list of all possible values of X together with their corresponding probabilities. We write P(X = x) or p(x) for the probability that X takes the value x.

离散随机变量的概率分布是X的所有可能取值及其对应概率的列表。我们用P(X = x)或p(x)表示X取值为x的概率。

For a valid probability distribution, two conditions must hold:

一个有效的概率分布必须满足两个条件：

Each probability is between 0 and 1 inclusive: 0 ≤ P(X = x) ≤ 1 for every x.
每个概率在0到1之间（含端点）：对每个x，0 ≤ P(X = x) ≤ 1。

The sum of all probabilities equals 1: Σ P(X = x) = 1 over all possible values x.
所有概率之和等于1：对所有可能取值x，Σ P(X = x) = 1。

0 ≤ p(x) ≤ 1 and Σₓ p(x) = 1

These two properties are the foundation for many exam questions that ask you to find an unknown value in a probability table.

这两个性质是许多考试题目的基础，这类题目要求你在概率表中求出未知值。

3. Expected Value E(X) | 期望值 E(X)

The expected value, also called the mean, is a measure of the central tendency of a discrete random variable. It is a weighted average of all possible values, where the weights are the corresponding probabilities.

期望值，也称为均值，是离散随机变量集中趋势的度量。它是所有可能取值的加权平均，其中权重为对应的概率。

E(X) = Σ x · P(X = x) = Σ x · p(x)

Here, the sum is taken over all possible values x of X. The expected value is denoted by μ (mu).

这里求和是对X的所有可能取值x进行的。期望值用μ表示。

Example: If X takes values 1, 2, 3 with probabilities 0.2, 0.3, 0.5, then E(X) = 1(0.2) + 2(0.3) + 3(0.5) = 0.2 + 0.6 + 1.5 = 2.3.
示例：若X取值1、2、3，概率分别为0.2、0.3、0.5，则E(X) = 1(0.2) + 2(0.3) + 3(0.5) = 0.2 + 0.6 + 1.5 = 2.3。

The expected value does not have to be a value that X can actually take. It is a long-run average in repeated trials.

期望值不一定是X实际能取的某个值。它是在重复试验中的长期平均值。

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

Variance measures the spread or dispersion of the distribution around the mean. It is the expected value of the squared deviation from the mean.

方差度量分布围绕均值的离散程度。它是偏离均值之平方的期望值。

Var(X) = E[(X − μ)²] = Σ (x − μ)² · p(x)

An equivalent and often more convenient formula is:

一个等价且通常更方便的公式是：

Var(X) = E(X²) − [E(X)]²

where E(X²) = Σ x² · p(x). The standard deviation is the square root of the variance: σ = √Var(X).

其中E(X²) = Σ x² · p(x)。标准差是方差的平方根：σ = √Var(X)。

Measure	Formula
Expected Value	μ = E(X) = Σ x · p(x)
E(X²)	E(X²) = Σ x² · p(x)
Variance	Var(X) = E(X²) − μ²
Standard Deviation	σ = √Var(X)

5. Properties of E(X) and Var(X) | E(X) 与 Var(X) 的性质

These linearity properties are crucial for solving problems involving transformations of random variables.

这些线性性质对于解决涉及随机变量变换的问题至关重要。

If X is a discrete random variable and a and b are constants, then:

若X是离散随机变量，a和b是常数，则：

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

Var(aX + b) = a² Var(X)

Note that adding a constant b shifts the mean but does not change the variance, while multiplying by a scales both the mean and the variance (variance scales by a²).

注意，加常数b会移动均值但不改变方差，而乘以a会同时缩放均值和方差（方差按a²缩放）。

Example: If E(X) = 5 and Var(X) = 4, then E(3X − 2) = 3(5) − 2 = 13, and Var(3X − 2) = 3² × 4 = 36.
示例：若E(X) = 5，Var(X) = 4，则E(3X − 2) = 3(5) − 2 = 13，Var(3X − 2) = 3² × 4 = 36。

6. Discrete Uniform Distribution | 离散均匀分布

A discrete random variable X has a uniform distribution if it takes each of its n possible values with equal probability. If X takes values x₁, x₂, …, xₙ, then P(X = xᵢ) = 1/n for each i.

如果离散随机变量X以等概率取n个可能值中的每一个，则称X服从均匀分布。若X取值为x₁, x₂, …, xₙ，则对每个i，P(X = xᵢ) = 1/n。

P(X = xᵢ) = 1/n for i = 1, 2, …, n

For a uniform distribution over integer values 1, 2, …, n, the mean and variance are:

对于在整数1, 2, …, n上的均匀分布，均值和方差为：

E(X) = (n + 1) / 2

Var(X) = (n² − 1) / 12

Example: Rolling a fair die gives a uniform distribution with n = 6. Thus E(X) = 7/2 = 3.5 and Var(X) = (36 − 1)/12 = 35/12 ≈ 2.917.
示例：掷公平骰子得到n = 6的均匀分布。因此E(X) = 7/2 = 3.5，Var(X) = (36 − 1)/12 = 35/12 ≈ 2.917。

7. Binomial Distribution | 二项分布

A binomial distribution models the number of successes in n independent Bernoulli trials, where each trial has the same probability of success p. We write X ~ B(n, p).

二项分布描述n次独立伯努利试验中成功次数的模型，其中每次试验的成功概率p相同。记作X ~ B(n, p)。

The probability of exactly k successes is given by:

恰好k次成功的概率为：

P(X = k) = ⁿCₖ · pᵏ · (1 − p)ⁿ⁻ᵏ

where ⁿCₖ = n! / [k!(n − k)!] is the binomial coefficient.

其中ⁿCₖ = n! / [k!(n − k)!]是二项式系数。

The mean and variance of a binomial distribution are:

二项分布的均值和方差为：

E(X) = np and Var(X) = np(1 − p)

Conditions for a binomial distribution: a fixed number n of independent trials, each with two outcomes (success/failure), and constant probability p of success.

二项分布的适用条件：固定的n次独立试验，每次试验只有两种结果（成功/失败），且成功概率p恒定。

8. Geometric Distribution | 几何分布

A geometric distribution models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials, each with probability of success p. We write X ~ Geo(p).

几何分布描述在独立伯努利试验序列中，直到首次成功所需的试验次数，每次试验的成功概率为p。记作X ~ Geo(p)。

P(X = k) = (1 − p)ᵏ⁻¹ · p for k = 1, 2, 3, …

The mean and variance of a geometric distribution are:

几何分布的均值和方差为：

E(X) = 1/p and Var(X) = (1 − p) / p²

For example, if p = 0.25, then E(X) = 4, meaning we expect 4 trials on average to get the first success.

例如，若p = 0.25，则E(X) = 4，意味着平均需要4次试验才能获得首次成功。

9. Poisson Distribution | 泊松分布

A Poisson distribution models the number of events occurring in a fixed interval of time or space, given that events occur independently at a constant average rate λ. We write X ~ Po(λ).

泊松分布描述在固定时间或空间间隔内事件发生次数的模型，条件是事件以恒定平均速率λ独立发生。记作X ~ Po(λ)。

P(X = k) = e⁻ᵡ · λᵏ / k! for k = 0, 1, 2, …

The mean and variance of a Poisson distribution are both equal to λ:

泊松分布的均值和方差都等于λ：

E(X) = λ and Var(X) = λ

A key feature of the Poisson distribution is that the variance equals the mean. This is a useful check in exam problems.

泊松分布的一个关键特征是方差等于均值。这是考试中一个有用的检验方法。

10. Combined Random Variables | 随机变量的组合

When dealing with two independent discrete random variables X and Y, we can find the expected value and variance of their sum or difference:

当处理两个独立的离散随机变量X和Y时，我们可以求其和或差的期望值和方差：

E(X + Y) = E(X) + E(Y)

E(X − Y) = E(X) − E(Y)

Var(X ± Y) = Var(X) + Var(Y)

Note that the variance of both the sum and the difference are equal; the minus sign does not change the variance formula because variance is always non-negative.

注意，和与差的方差相同；负号不会改变方差公式，因为方差总是非负的。

Example: If X ~ B(10, 0.3) and Y ~ B(20, 0.4) are independent, then E(X + Y) = 10(0.3) + 20(0.4) = 3 + 8 = 11.
示例：若X ~ B(10, 0.3)和Y ~ B(20, 0.4)独立，则E(X + Y) = 10(0.3) + 20(0.4) = 3 + 8 = 11。

11. Exam Strategies and Common Pitfalls | 考试策略与常见误区

IB exam questions on discrete random variables often require you to construct a probability distribution table, find the value of an unknown constant, compute E(X) and Var(X), and then apply a linear transformation or combine variables.

IB考试中关于离散随机变量的题目通常要求你构建概率分布表、求未知常数的值、计算E(X)和Var(X)，然后应用线性变换或组合变量。

Common pitfalls to avoid:

应避免的常见误区：

Forgetting that probabilities must sum to exactly 1. Always verify this before moving on.
忘记概率之和必须恰为1。在继续之前务必验证这一点。

Using Var(X) = E(X²) − μ² incorrectly: compute E(X²) first, then subtract the square of the mean, not the mean itself.
错误使用Var(X) = E(X²) − μ²：先计算E(X²)，再减去均值的平方，而不是均值本身。

Confusing the binomial coefficient ⁿCₖ with probability. The coefficient is part of the formula, not the probability itself.
混淆二项式系数ⁿCₖ与概率。系数是公式的一部分，不是概率本身。

Forgetting to check the conditions for using a particular distribution (e.g., independence for binomial, constant rate for Poisson).
忘记检查使用特定分布的条件（例如二项分布的独立性，泊松分布的恒定速率）。

When using a GDC (graphical display calculator), practice using the built-in distribution functions for binomial, geometric, and Poisson probabilities. Many IB questions allow calculator use, and knowing how to quickly compute P(X ≤ k) or P(X ≥ k) is a huge advantage.

使用图形计算器（GDC）时，练习使用内置的二项、几何和泊松概率分布函数。许多IB题目允许使用计算器，知道如何快速计算P(X ≤ k)或P(X ≥ k)是一个巨大优势。

12. Summary and Key Formulas | 总结与关键公式

The table below summarises the most important formulas for discrete random variables in IB Mathematics.

下表总结了IB数学中离散随机变量最重要的公式。

Concept	Formula
Probability Sum	Σ p(x) = 1
Expected Value	μ = Σ x · p(x)
E(X²)	Σ x² · p(x)
Variance	Var(X) = E(X²) − μ²
Linear Transformation	E(aX+b) = aE(X)+b; Var(aX+b) = a²Var(X)
Binomial: X ~ B(n, p)	P(X=k)=ⁿCₖpᵏ(1−p)ⁿ⁻ᵏ; E(X)=np; Var(X)=np(1−p)
Geometric: X ~ Geo(p)	P(X=k)=(1−p)ᵏ⁻¹p; E(X)=1/p; Var(X)=(1−p)/p²
Poisson: X ~ Po(λ)	P(X=k)=e⁻ᵡλᵏ/k!; E(X)=λ; Var(X)=λ

Mastering these formulas and understanding when to apply each distribution is the key to scoring top marks on discrete random variable questions in IB Mathematics. Practice with past papers and always double-check your probability tables before computing expected values.

掌握这些公式并理解何时应用每种分布，是在IB数学中离散随机变量题目取得高分的关键。用历年真题练习，并在计算期望值之前始终仔细检查你的概率表。

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IB Mathematics: Core Concepts of Discrete Random Variables | IB数学：离散随机变量核心概念