Mastering Expectation of Discrete Random Variables (Exercise 23C.1) | 掌握离散随机变量的期望（练习23C.1）

📚 Mastering Expectation of Discrete Random Variables (Exercise 23C.1) | 掌握离散随机变量的期望（练习23C.1）

In IB Mathematics Analysis and Approaches or Applications and Interpretation, understanding the expectation (mean) of a discrete random variable is a foundational skill. Exercise 23C.1 typically deals with calculating E(X), E(X²), and applying expectation rules. This article revisits the core concepts explored in this exercise, offering step-by-step techniques and common pitfalls to help you master the topic.

在IB数学分析与方法或应用与解释中，理解离散随机变量的期望（均值）是一项基础技能。练习23C.1通常涉及计算E(X)、E(X²)并应用期望的运算规则。本文将重新梳理该练习中的核心概念，提供逐步求解技巧和常见易错点，助你掌握这一主题。

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

A discrete random variable (X) takes a countable number of distinct values. Each value has an associated probability, and the sum of all probabilities must equal 1. Examples include the outcome of rolling a die (1 to 6) or the number of heads when flipping three coins. In Exercise 23C.1, you will often be given a probability distribution table and asked to compute expectations based on it.

离散随机变量 (X) 取有限或可数个不同的值。每个值都有相应的概率，且所有概率之和必须等于1。常见的例子包括掷骰子的结果（1到6）或掷三枚硬币时正面的次数。在练习23C.1中，通常会给出一个概率分布表，要求你据此计算期望值。

---

2. Probability Distributions Recap | 概率分布回顾

A probability distribution for a discrete random variable lists each possible outcome x and its probability P(X = x), often denoted as p(x). Two essential conditions must hold: 0 ≤ p(x) ≤ 1 for all x, and ∑ p(x) = 1. In tabular form, it looks like the following:

离散随机变量的概率分布列出了每个可能结果 x 及其概率 P(X = x)，通常记作 p(x)。必须满足两个基本条件：对所有 x 有 0 ≤ p(x) ≤ 1，且 ∑ p(x) = 1。用表格表示如下：

x	1	2	3	4
P(X=x)	0.1	0.2	0.3	0.4

Always verify that the probabilities sum to 1 before proceeding with any calculations; otherwise, the distribution is invalid.

在进行任何计算之前，务必验证这些概率之和为1；否则该分布是无效的。

---

3. The Concept of Expectation E(X) | 期望E(X)的概念

The expected value (or mean) of X, denoted E(X), represents the long-run average if the experiment is repeated many times. Mathematically, it is the probability-weighted average of all possible values:

期望值（或均值）E(X) 表示若此试验重复多次得到的长期平均值。在数学上，它是所有可能值的概率加权平均：

E(X) = ∑ xᵢ · p(xᵢ)

Think of E(X) as the ‘centre’ of the distribution. For a fair six-sided die, E(X) = 3.5. In Exercise 23C.1, you will practise computing this weighted sum directly from given tables.

可将 E(X) 视为分布的“中心”。对于一个公平的六面骰子，E(X) = 3.5。在练习23C.1中，你将练习直接从给定表格计算这一加权和。

---

4. Step-by-Step Calculation of E(X) from a Table | 从表格中逐步计算E(X)

To compute E(X), multiply each value x by its probability p(x), then add the products. For the distribution in Section 2:

计算 E(X) 时，将每个值 x 乘以其概率 p(x)，然后将这些乘积相加。以第2节的分布为例：

• 1 × 0.1 = 0.1
• 2 × 0.2 = 0.4
• 3 × 0.3 = 0.9
• 4 × 0.4 = 1.6

Summing these gives E(X) = 0.1 + 0.4 + 0.9 + 1.6 = 3.0. Always show your working clearly to minimise arithmetic errors. A second column for x · p(x) in your working table can be very helpful.

将这些乘积相加得到 E(X) = 0.1 + 0.4 + 0.9 + 1.6 = 3.0。务必清晰展示计算过程，以减少算术错误。在计算表格中添加 x · p(x) 这一列会很有帮助。

---

5. Understanding E(X²) and Its Importance | 理解E(X²)及其重要性

E(X²) is the expected value of the square of X. It is not the square of E(X), i.e. E(X²) ≠ [E(X)]² in general. Instead, you square the x-values first, then take the probability-weighted average:

E(X²) 是 X 平方的期望值。它并不是 E(X) 的平方，即一般情况下 E(X²) ≠ [E(X)]²。正确做法是先将每个 x 值平方，再求概率加权平均：

E(X²) = ∑ (xᵢ)² · p(xᵢ)

For our example: (1²×0.1) + (4×0.2) + (9×0.3) + (16×0.4) = 0.1 + 0.8 + 2.7 + 6.4 = 10.0. E(X²) later helps in computing variance, Var(X) = E(X²) − [E(X)]².

在我们的例子中：(1²×0.1) + (4×0.2) + (9×0.3) + (16×0.4) = 0.1 + 0.8 + 2.7 + 6.4 = 10.0。E(X²) 随后可用于计算方差 Var(X) = E(X²) − [E(X)]²。

---

6. Calculating E(g(X)) for a Function g | 计算函数g(X)的期望E(g(X))

If a new random variable is defined as a function of X, say Y = g(X), then its expectation is E(Y) = E(g(X)) = ∑ g(xᵢ) p(xᵢ). This is extremely useful in Exercise 23C.1 when you encounter expressions like 2X + 1 or X². You do not need to derive the whole distribution of Y; just apply the formula directly.

如果定义一个新随机变量为 X 的函数，例如 Y = g(X)，则其期望为 E(Y) = E(g(X)) = ∑ g(xᵢ) p(xᵢ)。这在练习23C.1中非常有用，当你遇到像 2X + 1 或 X² 这样的表达式时，无需推导 Y 的完整分布，直接套用公式即可。

For instance, to find E(2X + 1), compute (2×1+1)×0.1 + (2×2+1)×0.2 + (2×3+1)×0.3 + (2×4+1)×0.4, which equals (3×0.1) + (5×0.2) + (7×0.3) + (9×0.4) = 0.3 + 1.0 + 2.1 + 3.6 = 7.0.

例如，要求 E(2X + 1)，计算 (2×1+1)×0.1 + (2×2+1)×0.2 + (2×3+1)×0.3 + (2×4+1)×0.4，即 (3×0.1) + (5×0.2) + (7×0.3) + (9×0.4) = 0.3 + 1.0 + 2.1 + 3.6 = 7.0。

---

7. Linearity of Expectation: E(aX + b) | 期望的线性性质：E(aX + b)

A powerful shortcut is the linearity property: for constants a and b, E(aX + b) = aE(X) + b. This holds whether or not X is a discrete random variable. Using our earlier E(X) = 3.0, we immediately get E(2X + 1) = 2×3.0 + 1 = 7.0, which matches the direct calculation. This rule saves time and reduces calculation errors, especially for more complex linear transformations.

一个强大的捷径是线性性质：对于常数 a 和 b，有 E(aX + b) = aE(X) + b。无论 X 是否为离散随机变量，该性质均成立。利用之前得到的 E(X) = 3.0，可直接得到 E(2X + 1) = 2×3.0 + 1 = 7.0，与直接计算相符。此规则能节省时间并减少计算错误，尤其适用于更复杂的线性变换。

Note, however, that linearity does not extend to non-linear functions: E(X²) ≠ [E(X)]², and there is no simple formula for E(√X) in terms of E(X) alone.

但请注意，线性性质不适用于非线性函数：E(X²) ≠ [E(X)]²，且无法仅用 E(X) 表出 E(√X) 的简单公式。

---

8. Common Mistakes in Exercise 23C.1 | 练习23C.1中的常见错误

The most frequent errors arise from confusing E(X²) with (E(X))², or forgetting that probabilities must sum to 1 before computing expectations. Students also sometimes misread the table, applying the x-values to the wrong probabilities. To avoid these, always write a fresh working table with columns x, P(X=x), x·P, x²·P whenever you encounter a multi-step problem.

最常见的错误是将 E(X²) 与 [E(X)]² 混淆，或在计算期望前忘记检验概率之和为1。学生有时还会误读表格，将 x 值对应到错误的概率上。为避免这些问题，无论何时遇到多步问题，都应绘制一张包含 x、P(X=x)、x·P、x²·P 等列的工作表格。

Another pitfall is incorrectly assuming that E(aX + b) requires distributing the probability across new values individually, when linearity is much faster. Deliberate practice with mixed exercises will cement the correct approaches.

另一个易错点是错误地认为 E(aX + b) 需要将概率逐个分配到新值上，而实际上利用线性性质要快得多。通过混合练习有意识地训练，将有助于巩固正确的解法。

---

9. Practice Problem Walkthrough | 练习题目逐步解析

Let’s consolidate by solving a typical Exercise 23C.1 problem from scratch: A discrete random variable Y has probability distribution:

让我们通过从头求解一道典型的练习23C.1题目来巩固知识：离散随机变量 Y 的概率分布如下：

y	0	2	5	10
P(Y=y)	0.20	0.35	0.30	0.15

Step 1: Verify sum = 0.20+0.35+0.30+0.15 = 1.00. ✓

第1步：验证和 = 0.20+0.35+0.30+0.15 = 1.00。 ✓

Step 2: Find E(Y). E(Y) = 0×0.20 + 2×0.35 + 5×0.30 + 10×0.15 = 0 + 0.70 + 1.50 + 1.50 = 3.70.

第2步：求 E(Y)。E(Y) = 0×0.20 + 2×0.35 + 5×0.30 + 10×0.15 = 0 + 0.70 + 1.50 + 1.50 = 3.70。

Step 3: Find E(Y²). E(Y²) = 0²×0.20 + 2²×0.35 + 5²×0.30 + 10²×0.15 = 0 + 4×0.35 + 25×0.30 + 100×0.15 = 1.40 + 7.50 + 15.00 = 23.90.

第3步：求 E(Y²)。E(Y²) = 0²×0.20 + 2²×0.35 + 5²×0.30 + 10²×0.15 = 0 + 4×0.35 + 25×0.30 + 100×0.15 = 1.40 + 7.50 + 15.00 = 23.90。

Step 4: If W = 3Y − 5, find E(W). Using linearity: E(W) = 3E(Y) − 5 = 3×3.70 − 5 = 11.10 − 5 = 6.10.

第4步：若 W = 3Y − 5，求 E(W)。利用线性性质：E(W) = 3E(Y) − 5 = 3×3.70 − 5 = 11.10 − 5 = 6.10。

Direct verification: E(3Y−5) = ( −5)×0.20 + (1)×0.35 + (10)×0.30 + (25)×0.15 = −1.0 + 0.35 + 3.0 + 3.75 = 6.10. Perfect agreement.

直接验证：E(3Y−5) = ( −5)×0.20 + (1)×0.35 + (10)×0.30 + (25)×0.15 = −1.0 + 0.35 + 3.0 + 3.75 = 6.10。完全吻合。

---

10. Key Takeaways and Exam Tips | 关键要点与考试贴士

Mastering Exercise 23C.1 builds a strong foundation for expected value problems across the IB syllabus. Always remember: (1) E(X) is the probability-weighted average; (2) E(X²) squares the x-values first; (3) E(aX + b) = aE(X) + b; and (4) never assume non-linear relationships simplify similarly. In an exam, you may be asked to compute expectation to then find variance or apply it to insurance premiums, game fairness, etc. For complex distributions, a well-organised table will be your best tool.

掌握练习23C.1将为整个IB课程中的期望值问题打下坚实基础。务必牢记：（1）E(X) 是概率加权平均；（2）E(X²) 先对 x 值平方；（3）E(aX + b) = aE(X) + b；（4）切勿假设非线性关系可以类似简化。考试中可能要求你计算期望，进而求方差或将其应用于保险费、游戏公平性等问题。对于复杂的分布，一张条理清晰的表格将是你最好的工具。

Finally, practise with a variety of probability tables and transformations until the steps become automatic. The more you work with exercises like 23C.1, the more confident you will become in the statistics portion of IB Mathematics.

最后，通过多种概率分布表及变换进行练习，直到这些步骤变为本能。类似23C.1这样的练习做得越多，你在IB数学统计部分就会越自信。

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）