📚 Review Set 19C | 复习题集19C
Review Set 19C in the IB Mathematics course focuses on consolidating concepts from the Probability Distributions chapter. It brings together discrete random variables, expectation, variance, binomial, Poisson, and normal distributions, challenging students to apply analytical skills and calculator techniques in typical IB exam-style problems. This article provides a thorough walkthrough of the key ideas, formulas, and common pitfalls you will encounter in this review set.
IB 数学课程中的复习题集 19C 集中巩固了“概率分布”一章的核心概念。它将离散随机变量、期望、方差、二项分布、泊松分布以及正态分布融合在一起，要求学生运用分析能力和计算器技巧应对典型的 IB 考试题型。本文将对你在该题集中会遇到的关键思想、公式和常见误区进行全面解析。
1. Understanding Discrete Random Variables | 理解离散随机变量
A discrete random variable X takes on a finite or countably infinite set of values, each with an associated probability. In Review Set 19C, you will often need to verify that the sum of all probabilities equals 1, and that each probability lies between 0 and 1.
离散随机变量 X 取有限或可数无限个值，每个值都有相应的概率。在复习题集 19C 中，你经常需要验证所有概率之和为 1，并且每个概率都在 0 到 1 之间。
te probabilities like P(X > k) or P(a ≤ X < b). Remember that P(X = a) is denoted as p(a).
例如，概率质量函数可能以表格形式给出。你需要阅读表格，求未知常数，并计算诸如 P(X > k) 或 P(a ≤ X < b) 的概率。请记住 P(X = a) 记作 p(a)。
2. Constructing Probability Distribution Tables | 构建概率分布表
A typical question in Review Set 19C asks you to set up a probability distribution from a word problem. You might need to list all outcomes of an experiment and assign probabilities using basic rules such as equally likely outcomes or tree diagrams.
复习题集 19C 中的典型题目会要求你根据文字题建立概率分布。你可能需要列出实验的所有结果，并使用等可能结果或树状图等基本规则分配概率。
For instance, if p(x) = kx for x = 1,2,3, then 1k + 2k + 3k = 1 yields 6k = 1, so k = 1/6.
务必检查表格的有效性：∑p(x) = 1。如果含有未知参数，利用这一条件求出它。例如，若 p(x) = kx，x = 1,2,3，则 k + 2k + 3k = 1，解得 6k = 1，因此 k = 1/6。
3. Expected Value E(X) | 期望值 E(X)
lems, you will often compute E(X) directly from a table and then use it in further decision-making contexts.
期望值是随机变量长期运行的平均值，计算公式为 E(X) = Σ x · P(X = x)。在 IB 试题中，你通常会直接从表格计算 E(X)，然后将其用于进一步的决策情境。
If the distribution is symmetric, the mean is at the centre, but for skewed distributions, the expected value is pulled toward the tail. Make sure you multiply each x-value by its exact probability before summing.
如果分布是对称的，均值位于中心，但对于偏态分布，期望值会偏向尾部。在求和之前，务必确保将每个 x 值与其精确概率相乘。
4. Variance and Standard Deviation | 方差与标准差
ou are expected to be comfortable using the computational formula Var(X) = Σ x² p(x) − μ².
方差衡量分布的离散程度，其定义为 Var(X) = E[(X − μ)²] = E(X²) − [E(X)]²。在复习题集 19C 中，要求你熟练掌握计算式 Var(X) = Σ x² p(x) − μ²。
n an IB exam, you may be asked to interpret σ in context, or to use it when standardizing for the normal distribution.
标准差是方差的正平方根：σ = √Var(X)。在 IB 考试中，你可能需要解释 σ 的实际意义，或者在正态分布标准化时用到它。
5. The Binomial Distribution B(n, p) | 二项分布 B(n, p)
The binomial distribution models the number of successes in n independent trials, each with constant success probability p. A random variable X ~ B(n, p) has probability mass function:
二项分布描述在 n 次独立试验中成功的次数，每次成功概率恒为 p。随机变量 X ~ B(n, p) 的概率质量函数为：
P(X = k) = nCk pk (1 − p)n − k
our GDC. Typical Review Set 19C problems include finding P(X = r), P(X ≤ r), and P(X > r).
你需要既能够手动计算（对于较小的 n），又能用图形计算器计算二项概率。复习题集 19C 中的典型问题包括求 P(X = r)、P(X ≤ r) 和 P(X > r)。
p). These often appear in theoretical parts of questions where you have to deduce n or p.
请记住二项分布的均值 μ = np，方差 σ² = np(1 − p)。这些经常出现在需要推断 n 或 p 的理论性质题目中。
6. The Poisson Distribution Po(λ) | 泊松分布 Po(λ)
The Poisson distribution models the number of events occurring in a fixed interval of time or space, provided events happen independently at a constant average rate λ. Its probability mass function is:
泊松分布模型用于描述在固定时间或空间区间内发生的事件次数，条件是事件以恒定平均速率 λ 独立发生。其概率质量函数为：
P(X = k) = (λk e−λ) / k!
In Review Set 19C, you'll apply the Poisson distribution to contexts like call-centre calls, defects in a roll of cloth, or radioactive decay. You often need to add independent Poisson variables: if X ~ Po(λ₁) and Y ~ Po(λ₂) are independent, then X + Y ~ Po(λ₁ + λ₂).
在复习题集 19C 中，你将会把泊松分布应用于诸如呼叫中心来电、布匹瑕疵或放射性衰变等问题。你经常需要相加独立泊松变量：若 X ~ Po(λ₁) 与 Y ~ Po(λ₂) 独立，则 X + Y ~ Po(λ₁ + λ₂)。
Both mean and variance of a Poisson random variable equal λ. This property is frequently tested when you have to decide whether a distribution is Poisson based on sample data.
泊松随机变量的均值和方差都等于 λ。当你需要根据样本数据判断某一分布是否为泊松分布时，这一性质常常会被考查。
7. The Normal Distribution and Standardisation | 正态分布与标准化
The normal distribution X ~ b) directly, or use the standard normal variable Z = (X − μ) / σ when working with tables.
正态分布 X ~ N(μ, σ²) 是连续且对称的。你将使用图形计算器直接求概率 P(a < X < b)，或者在查表时使用标准正态变量 Z = (X − μ) / σ。
Z = (X − μ) / σ
en probability. These are common in Review Set 19C and require careful handling of tail areas.
逆正态分布问题要求找出满足 P(X ≤ x) = 给定概率的 x 值。这类题在复习题集 19C 中很常见，需要仔细处理尾部的面积。
You may also need to combine a binomial approximation with a normal distribution: if n is large and p is close to 0.5, X ~ B(n, p) can be approximated by N(np, np(1 − p)). The continuity correction is rarely used in IB, but recognising the approximation could be required.
你或许还需要将二项分布近似为正态分布：如果 n 很大且 p 接近 0.5，X ~ B(n, p) 可用 N(np, np(1 − p)) 近似。尽管连续性校正很少在 IB 中使用，但识别这种近似可能是必要的。
8. Using GDC for Probability Distributions | 使用图形计算器处理概率分布
For normal, normalcdf(lower, upper, μ, σ) and invNorm(area, μ, σ) are your primary tools.
IB 要求高效使用图形计算器。对于二项与泊松分布，你可以使用内置分布函数：binompdf(n, p, k) 给出 P(X = k)，binomcdf(n, p, k) 给出 P(X ≤ k)。对于正态分布，normalcdf(下界, 上界, μ, σ) 和 invNorm(面积, μ, σ) 是你的主要工具。
In Review Set 19C, you may need to interpret GDC output and write down probabilities correct to three or four decimal places. Always sketch a diagram for normal probability problems to avoid confusion between lower and upper tails.
在复习题集 19C 中，你可能需要解读计算器输出并以三位或四位小数写下概率。对于正态概率问题，始终画一个简图，以避免混淆下尾和上尾。
9. Worked IB-Style Problems from Review Set 19C | 来自复习题集19C的IB风格例题
ir die is rolled 12 times. Find the probability that a six appears exactly 4 times. Here X ~ B(12, 1/6). Using GDC, binompdf(12, 1/6, 4) ≈ 0.0888. The manual computation would use 12C4 (1/6)4 (5/6)8.
考虑这个问题：一个公平的骰子掷 12 次。求正好出现 4 次六点的概率。这里 X ~ B(12, 1/6)。使用计算器，binompdf(12, 1/6, 4) ≈ 0.0888。手动计算则用 12C4 (1/6)4 (5/6)8。
the probability that a 3-metre length has at most 1 defect. Here λ for 3 metres = 6. So X ~ Po(6). P(X ≤ 1) = P(X=0) + P(X=1) = e⁻⁶ + 6e⁻⁶ = 7e⁻⁶ ≈ 0.0174.
另一个典型题：织物上的瑕疵随机发生，平均每米 2 个。求 3 米布料上最多有 1 个瑕疵的概率。这里 3 米的 λ = 6，所以 X ~ Po(6)。P(X ≤ 1) = P(X=0) + P(X=1) = e⁻⁶ + 6e⁻⁶ = 7e⁻⁶ ≈ 0.0174。
probability that a randomly chosen apple weighs between 140 g and 165 g. Using normalcdf(140, 165, 150, 20) gives about 0.4649.
正态分布问题：苹果的重量服从正态分布，μ = 150 g，σ = 20 g。求随机选取的一个苹果重量在 140 g 到 165 g 之间的概率。用 normalcdf(140, 165, 150, 20) 计算得大约 0.4649。
10. Common Pitfalls and Exam Tips | 常见易错点与考试技巧
One frequent mistake is confusing the binomial and Poisson distributions. Binomial has a fixed number of trials n, while Poisson rarely specifies n and focuses on the rate. Always check whether the scenario involves a count of successes out of a fixed total, or simply events in an interval.
一个常见错误是混淆二项分布与泊松分布。二项分布有固定的试验次数 n，而泊松分布很少指定 n，关注的是速率。务必检查情境涉及的是在固定总次数下的成功计数，还是仅仅间隔内的事件数。
For binomial, use directly σ² = np(1 − p) instead of struggling with the general formula.
使用方差公式时，许多学生忘记减去均值的平方。仔细计算 E(X²)：先将 x 平方，再乘以 p(x)，求和，最后减去 μ²。对于二项分布，直接使用 σ² = np(1 − p)，而无需费力用一般公式。
Always check 'calculator notation' in your working. Even when using GDC functions, write down the distribution type and parameters. For normal probabilities, draw a bell curve and shade the area. This helps you set up the lower and upper bounds correctly, especially for right-tail problems.
在解题过程中，务必写出“计算器符号”。即使使用计算器函数，也要写出分布类型与参数。对于正态概率，画出钟形曲线并标出阴影面积。这有助于你正确设定下界和上界，尤其是对于右尾问题。
11. Connections to the Broader IB Syllabus | 与IB更广泛考纲的联系
The probability distributions in Review Set 19C lay the foundation for statistical hypothesis testing, which appears later in the course. Understanding binomial and Poisson distributions is crucial for tests concerning proportions and rates of occurrence.
复习题集 19C 中的概率分布为后续课程中的统计假设检验奠定基础。理解二项分布和泊松分布对于关于比例和发生率参数的检验至关重要。
Moreover, the normal distribution connects directly to confidence intervals and error analysis in IA projects. The fluency gained with GDC functions here will save you time and reduce errors in Paper 2 and your internal assessment.
此外，正态分布直接与内部评估（IA）项目中的置信区间和误差分析相连。熟练掌握这里的图形计算器函数将为你节省时间，减少试卷二以及内部评估中的错误。
12. Final Summary of Key Formulas | 关键公式最终小结
| Distribution | Notation | Mean | Variance | Probability Function |
|---|---|---|---|---|
| Binomial | X ~ B(n, p) | np | np(1−p) | nCk pk (1−p)n−k |
| Poisson | X ~ Po(λ) | λ | λ | (λk e−λ) / k! |
| Normal | X ~ N(μ, σ²) | μ | σ² | f(x) = (1/(σ√2π)) e−(x−μ)²/(2σ²) |
Keep this table in mind as you work through Review Set 19C; it distils the essential parameters you must recall in an exam. Practise switching between the formula and GDC commands until the process becomes second nature.
在你做复习题集 19C 时，请牢记这张表格；它浓缩了你在考试中必须回忆的关键参数。反复练习在公式和计算器命令之间切换，直到整个过程成为第二天性。
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