IB Mathematics: Discrete Random Variables Exam Guide | IB数学：离散随机变量考点梳理

📚 IB Mathematics: Discrete Random Variables Exam Guide | IB数学：离散随机变量考点梳理

This guide covers all the key ideas you need for discrete random variables in IB Mathematics. You will learn how to define a discrete random variable, write its probability distribution, calculate expectation and variance, apply linear transformations, and solve binomial distribution questions.

本文为你系统梳理IB数学中离散随机变量的核心考点，包括概率分布的定义、期望与方差的计算、线性变换、二项分布，以及考试中常见的易错点，帮助你在考场上快速准确作答。

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

A random variable is a numerical value determined by the outcome of a random experiment. It is called discrete when it can take only a countable set of values, such as 0, 1, 2, 3, … . These values can usually be listed individually, and each value has a probability assigned to it.

随机变量是随机试验结果所对应的数值。若其可能的取值是有限个或可数无穷多个，我们就称它为离散随机变量。例如抛硬币出现的正面次数、掷骰子得到的点数、一批产品中抽到次品的数量等，都是离散随机变量。

2. Probability Mass Function and Its Conditions | 概率分布与概率质量函数

The probability distribution of a discrete random variable \(X\) is usually described by its probability mass function, written as \(P(X = x)\), or simply \(p(x)\). This function gives the probability that the random variable takes a particular value.

离散随机变量 \(X\) 的概率分布通常用概率质量函数表示，记作 \(P(X = x)\) 或 \(p(x)\)。它表示 \(X\) 取某个特定值时的概率。

0 ≤ P(X = x) ≤ 1 for every possible value x.
每个可能取值 x 的概率满足 0 ≤ P(X = x) ≤ 1。
The sum of all probabilities must equal 1.
所有概率之和必须等于 1。

For example, if X can take values 1, 2 and 3 only, we might have:

例如，若 X 只可能取 1、2、3，则其概率分布可表示为：

x	1	2	3
P(X = x)	0.2	0.3	0.5

Notice that 0.2 + 0.3 + 0.5 = 1, so this is a valid probability distribution.

注意 0.2 + 0.3 + 0.5 = 1，因此这是一个合法的概率分布。

3. Cumulative Distribution Function | 累积分布函数

The cumulative distribution function, or CDF, gives the probability that the random variable is less than or equal to a certain value. It is usually written as F(x) = P(X ≤ x).

累积分布函数表示随机变量小于或等于某个值的概率，通常记作 F(x) = P(X ≤ x)。

F(x) = P(X ≤ x) = sum of P(X = t) for all t ≤ x

Using the example above, F(2) = P(X = 1) + P(X = 2) = 0.2 + 0.3 = 0.5. The cumulative distribution function is non-decreasing and always approaches 1 as x increases.

用上面的例子计算：F(2) = P(X = 1) + P(X = 2) = 0.2 + 0.3 = 0.5。累积分布函数是非递减的，当 x 增大时，它的值最终趋近于 1。

4. Expected Value E(X) | 期望 E(X)

The expected value of a discrete random variable is the weighted average of all possible values, using the probabilities as weights. It is denoted by E(X) or μ (mu).

离散随机变量的期望是所有可能取值的加权平均值，权重就是对应的概率。它通常记作 E(X) 或 μ。

E(X) = ∑ xᵢ P(X = xᵢ)

For the earlier distribution, E(X) = 1(0.2) + 2(0.3) + 3(0.5) = 0.2 + 0.6 + 1.5 = 2.3. This means that if the experiment is repeated many times, the long-run average value is 2.3.

对于之前给出的分布，E(X) = 1(0.2) + 2(0.3) + 3(0.5) = 0.2 + 0.6 + 1.5 = 2.3。这意味着如果试验重复很多次，长期平均结果约为 2.3。

Remember: the expected value does not have to be one of the possible values of X.

注意：期望值并不一定是随机变量实际可能取得的某个值。

5. Variance and Standard Deviation | 方差和标准差

Variance measures how spread out the possible values are around the mean. A large variance means the values are widely spread, and a small variance means they are concentrated near the mean.

方差衡量随机变量的取值相对于期望的分散程度。方差越大，说明取值越分散；方差越小，说明取值越集中在均值附近。

Var(X) = ∑ (xᵢ – μ)² P(X = xᵢ)

In IB exams, the computational formula is often quicker to use:

在IB考试中，计算时通常使用下面的公式会更加快捷：

Var(X) = E(X²) – [E(X)]²

Here E(X²) = ∑ xᵢ² P(X = xᵢ). For the example above:

其中 E(X²) = ∑ xᵢ² P(X = xᵢ)。以上面的例子计算：

E(X²) = 1²(0.2) + 2²(0.3) + 3²(0.5) = 0.2 + 1.2 + 4.5 = 5.9

Therefore Var(X) = 5.9 – 2.3² = 5.9 – 5.29 = 0.61. The standard deviation is σ = √Var(X).

因此 Var(X) = 5.9 – 2.3² = 5.9 – 5.29 = 0.61。标准差为 σ = √Var(X)。

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

If X is a discrete random variable and a and b are constants, then the new variable Y = aX + b is also a discrete random variable. You must know how expectation and variance change under this transformation.

若 X 是离散随机变量，a 和 b 是常数，那么新变量 Y = aX + b 仍然是离散随机变量。你需要掌握期望和方差在线性变换下的变化规律。

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

Var(aX + b) = a² Var(X)

Adding a constant shifts all values but does not change the spread, so the variance is unchanged by adding b. Multiplying by a constant does change the spread, and the variance is multiplied by a², not a.

常数 b 只改变整体位置，不改变数据的离散程度，因此方差不变；而常数 a 会改变分散程度，方差要乘以 a²，而不是 a。

7. Mode and Median of a Discrete Random Variable | 众数和中位数

The mode is the value of X that has the highest probability. It is the most likely single outcome.

众数是概率最大的取值，也就是最可能出现的单个结果。

The median is the smallest value x such that P(X ≤ x) ≥ 0.5. This is the middle value of the distribution when probabilities are accumulated.

中位数是使 P(X ≤ x) ≥ 0.5 成立的最小取值 x，也就是概率累积到一半时对应的值。

For the example earlier, P(X = 3) = 0.5, so the mode is 3. Also F(2) = 0.5, so the median is 2 because 2 is the smallest value with cumulative probability at least 0.5.

对之前的例子而言，P(X = 3) = 0.5，因此众数是 3；同时 F(2) = 0.5，使得累积概率至少为 0.5 的最小值是 2，因此中位数是 2。

8. The Binomial Distribution | 二项分布

The binomial distribution is one of the most important discrete distributions in IB Mathematics. Let X be the number of successes in n independent trials, where each trial has two possible outcomes: success or failure, and success has probability p.

二项分布是IB数学中最重要的离散分布之一。设 X 是 n 次独立试验中“成功”的次数，每次试验只有成功或失败两种结果，且成功的概率为 p。

The key conditions for a binomial random variable are:

二项随机变量的关键条件如下：

There is a fixed number of trials, n.
试验次数 n 固定。
Each trial has only two outcomes: success and failure.
每次试验只有成功和失败两种结果。
The trials are independent.
各次试验相互独立。
The probability of success, p, is constant for every trial.
每次试验成功的概率 p 保持不变。

We write X ~ B(n, p). The probability of exactly r successes is:

我们记作 X ~ B(n, p)。恰好出现 r 次成功的概率为：

P(X = r) = C(n,r) pʳ (1-p)ⁿ⁻ʳ

The mean and variance of a binomial random variable are given by:

二项随机变量的期望和方差为：

E(X) = np

Var(X) = np(1-p)

For example, if X ~ B(10, 0.3), then E(X) = 10 × 0.3 = 3 and Var(X) = 10 × 0.3 × 0.7 = 2.1.

例如，若 X ~ B(10, 0.3)，则 E(X) = 10 × 0.3 = 3，Var(X) = 10 × 0.3 × 0.7 = 2.1。

9. Binomial Probabilities on Your GDC | 使用图形计算器求二项分布概率

In an IB exam, you are expected to use your GDC to calculate binomial probabilities quickly. Make sure you are familiar with the distribution menu of your calculator model.

在IB考试中，你需要使用图形计算器快速计算二项分布概率，因此务必熟悉自己计算器的分布菜单。

On TI calculators, use binompdf(n, p, r) to find P(X = r).
在 TI 计算器中，使用 binompdf(n, p, r) 求 P(X = r)。
Use binomcdf(n, p, r) to find P(X ≤ r).
使用 binomcdf(n, p, r) 求 P(X ≤ r)。
For P(X < r), use P(X ≤ r - 1) or binomcdf(n, p, r - 1).
求 P(X < r) 时，可转化为 P(X ≤ r - 1)，即 binomcdf(n, p, r - 1)。
For P(X ≥ r), use 1 – P(X ≤ r – 1).
求 P(X ≥ r) 时，使用 1 – P(X ≤ r – 1)。

Always write down the probability expression clearly before giving the calculator answer. This shows the examiner your method.

在给出计算器答案前，先清晰地写出概率表达式，这能向考官展示你的解题思路。

10. Worked Example: Putting It All Together | 综合例题：综合运用

A discrete random variable X has the following probability distribution:

已知离散随机变量 X 的概率分布如下：

x	1	2	3
P(X = x)	0.2	0.3	0.5

(a) Show that this is a valid probability distribution.

（a）证明这是一个合法的概率分布。

All probabilities are between 0 and 1, and 0.2 + 0.3 + 0.5 = 1, so the distribution is valid.

所有概率都在 0 和 1 之间，且 0.2 + 0.3 + 0.5 = 1，因此该分布合法。

(b) Find E(X).

（b）求 E(X)。

E(X) = 1(0.2) + 2(0.3) + 3(0.5) = 2.3

（c）求 Var(X)。

E(X²) = 1²(0.2) + 2²(0.3) + 3²(0.5) = 5.9

Var(X) = 5.9 – 2.3² = 0.61

(d) Find E(4X – 2) and Var(4X – 2).

（d）求 E(4X –

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IB Mathematics: Discrete Random Variables Exam Guide | IB数学：离散随机变量考点梳理