Exercise 21E.2: Poisson Distribution | 练习21E.2：泊松分布

📚 Exercise 21E.2: Poisson Distribution | 练习21E.2：泊松分布

Exercise 21E.2 is designed to build your confidence in applying the Poisson distribution to a wide range of probability problems. It covers the core probability formula, the use of calculator functions, cumulative probabilities, sums of independent Poisson variables, and the Poisson approximation to the binomial distribution. Each question challenges you to move from abstract parameters to concrete real-world contexts, such as call centres, traffic flow and defect rates.

练习 21E.2 旨在帮助你熟练运用泊松分布解决各种概率问题。它涵盖了核心概率公式、计算器功能的使用、累积概率、独立泊松变量之和，以及用泊松分布近似二项分布。每个问题都要求你将抽象的参数与具体的现实场景（如呼叫中心、交通流量和缺陷率）联系起来。

1. Definition and Conditions | 定义与条件

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, provided these events happen with a known constant mean rate and independently of the time since the last event. A random variable X follows a Poisson distribution with parameter λ (lambda) if its possible values are 0, 1, 2, 3, … and λ > 0.

泊松分布是一种离散概率分布，它表示在固定时间或空间间隔内发生某一给定数量事件的概率，前提是这些事件以已知的恒定平均速率发生，且与上一次事件发生的时间无关。如果一个随机变量 X 的可能取值为 0、1、2、3……，且参数 λ（拉姆达）> 0，则称 X 服从参数为 λ 的泊松分布。

The three main conditions for a Poisson model are: (1) events occur singly – two events cannot happen at exactly the same instant; (2) events occur independently – one event does not influence the probability of another; (3) the average rate at which events occur is constant throughout the interval.

泊松模型需要满足三个主要条件：(1) 事件单个发生——两个事件不能在同一瞬间发生；(2) 事件独立发生——一个事件不影响另一个事件发生的概率；(3) 事件发生的平均速率在整个区间内保持恒定。

2. Probability Mass Function | 概率质量函数

If X ~ Po(λ), the probability that X takes a particular integer value k is given by the formula:

若 X ~ Po(λ)，则 X 取某个整数 k 的概率由下式给出：

P(X = k) = (λᵏ e⁻λ) / k! , k = 0, 1, 2, 3, …

Here e is Euler’s number (approximately 2.71828) and k! denotes the factorial of k. This formula is the foundation of every question in Exercise 21E.2. You will often need to use it manually for small values of k or when a calculator use is restricted, but knowing how λ and k affect the shape of the distribution is equally important.

其中 e 是自然常数（约为 2.71828），k! 表示 k 的阶乘。这一公式是练习 21E.2 中所有问题的基石。当 k 值较小时，或当计算器使用受限时，你经常需要手动代入公式，但了解 λ 和 k 如何影响分布的形状同样重要。

3. Mean and Variance of a Poisson Distribution | 泊松分布的均值与方差

A remarkable property of the Poisson distribution is that its mean and its variance are both equal to λ. That is, E(X) = Var(X) = λ. This property makes it very easy to check whether a given data set can be modelled by a Poisson distribution: simply calculate the sample mean and sample variance – if they are roughly equal, a Poisson model may be appropriate.

泊松分布的一个显著性质是，它的均值和方差都等于 λ。也就是说，E(X) = Var(X) = λ。这一性质使得检查一组数据是否能用泊松分布建模变得非常容易：只需计算样本均值和样本方差，如果两者大致相等，那么泊松模型可能是合适的。

In Exercise 21E.2, you may be asked to deduce λ from a word problem (e.g. ‘the mean number of calls per minute is 4.2’) and then immediately use it in probability calculations. Remember that the standard deviation is √λ, which also appears in normal approximations for large λ.

在练习 21E.2 中，你可能会被要求从文字题中推断出 λ（例如“每分钟的平均呼叫次数为 4.2”），然后立即将其用于概率计算。请记住标准差为 √λ，当 λ 较大时，它也会出现在正态近似中。

4. Using the Calculator: pdf and cdf | 使用计算器：pdf 与 cdf

For IB examinations, you are expected to use a graphical calculator efficiently. The Poisson probability mass function is accessed via the poissonpdf(λ, k) function, which computes P(X = k). For cumulative probabilities P(X ≤ k), use poissoncdf(λ, k). Many questions in Exercise 21E.2 ask for ‘at least’, ‘more than’ or ‘between’ values, which you must express in terms of ≤ before entering them into the cdf function.

在 IB 考试中，你需要熟练使用图形计算器。泊松概率质量函数通过 poissonpdf(λ, k) 调用，它可以计算 P(X = k)。对于累积概率 P(X ≤ k)，请使用 poissoncdf(λ, k)。练习 21E.2 中的许多题目都会要求计算“至少”“多于”或“介于”某值之间的概率，你必须先将其转换为 ≤ 的形式，再输入 cdf 函数。

For example, P(X > 5) = 1 – P(X ≤ 5). P(3 ≤ X ≤ 7) = P(X ≤ 7) – P(X ≤ 2). Always write down the transformed expression before reaching for the calculator – this reduces careless errors.

例如，P(X > 5) = 1 – P(X ≤ 5)；P(3 ≤ X ≤ 7) = P(X ≤ 7) – P(X ≤ 2)。在拿起计算器之前，一定要先写下转换后的表达式，这样可以减少粗心错误。

5. Worked Example 1: Basic Exact Probability | 例题1：基本精确概率

Question: A receptionist receives on average 3.5 emergency calls per hour. Find the probability that in a randomly chosen hour she receives exactly 2 emergency calls. Let X ~ Po(3.5). Then P(X = 2) = (3.5² × e⁻³·⁵) / 2! = (12.25 × 0.030197) / 2 ≈ 0.1850. Using poissonpdf(3.5, 2) gives the same value.

题目：一名接待员平均每小时接到 3.5 通紧急电话。求在随机选择的一个小时内她恰好接到 2 通紧急电话的概率。设 X ~ Po(3.5)，则 P(X = 2) = (3.5² × e⁻³·⁵) / 2! = (12.25 × 0.030197) / 2 ≈ 0.1850。使用 poissonpdf(3.5, 2) 会得到相同的结果。

Always state the random variable, the distribution, and the exact expression before rounding your final answer to three significant figures unless instructed otherwise. Exercise 21E.2 typically asks for answers correct to three significant figures.

在将最终答案四舍五入至三位有效数字（除非另有说明）之前，一定要先写出随机变量、分布和准确的表达式。练习 21E.2 通常会要求将答案保留到三位有效数字。

6. Worked Example 2: Cumulative Probability | 例题2：累积概率

Question: The number of cars passing a checkpoint follows a Poisson distribution with a mean of 4.2 cars per minute. Find the probability that in a given minute more than 5 cars pass the checkpoint. Let Y ~ Po(4.2). We want P(Y > 5) = 1 – P(Y ≤ 5). Using poissoncdf(4.2, 5) yields P(Y ≤ 5) ≈ 0.746, so P(Y > 5) ≈ 0.254.

题目：经过某一检查站的汽车数量服从泊松分布，平均每分钟 4.2 辆。求在某一分钟内超过 5 辆车经过检查站的概率。设 Y ~ Po(4.2)。需要求 P(Y > 5) = 1 – P(Y ≤ 5)。利用 poissoncdf(4.2, 5) 可得 P(Y ≤ 5) ≈ 0.746，因此 P(Y > 5) ≈ 0.254。

When the question uses the phrasing ‘more than’, ‘at least’, ‘no more than’ or ‘fewer than’, always translate carefully. ‘At least 3’ means X ≥ 3 = 1 – P(X ≤ 2). ‘Fewer than 4’ means X < 4 = P(X ≤ 3). Such translations are frequently tested in Exercise 21E.2.

当题目中出现“多于”“至少”“不多于”或“少于”等表述时，一定要仔细转换。“至少 3 个”意味着 X ≥ 3 = 1 – P(X ≤ 2)。“少于 4 个”意味着 X < 4 = P(X ≤ 3)。这类转换在练习 21E.2 中常常出现。

7. Sum of Independent Poisson Variables | 独立泊松变量之和

If X ~ Po(λ₁) and Y ~ Po(λ₂) are independent random variables, then their sum X + Y follows a Poisson distribution with parameter λ₁ + λ₂. This property is extremely useful when combining counts from two independent sources. For example, if phone calls arrive on line A at a rate of 3 per minute and on line B at a rate of 2 per minute, the total calls per minute on both lines is Po(5).

如果 X ~ Po(λ₁) 和 Y ~ Po(λ₂) 是相互独立的随机变量，那么它们的和 X + Y 服从参数为 λ₁ + λ₂ 的泊松分布。在合并两个独立来源的计数时，这一性质非常有用。例如，若电话线路 A 每分钟来电平均 3 通，线路 B 每分钟平均 2 通，则两条线路每分钟的总来电次数服从 Po(5)。

Exercise 21E.2 often includes problems where you must first establish that two Poisson processes are independent and then compute probabilities for the total. Do not attempt to add probabilities directly – add the rates and then use the new λ for the combined distribution.

练习 21E.2 常常包含这样一些问题：你必须首先确认两个泊松过程是独立的，然后计算总和的概率。不要直接对概率进行相加，而应先将速率相加，再使用新的 λ 进行运算。

8. Poisson Approximation to the Binomial | 用泊松分布近似二项分布

When n is large and p is small, the binomial distribution B(n, p) can be approximated by a Poisson distribution with λ = np. The approximation is considered good when n > 50 and np < 5, or when n > 20 and p < 0.1. In Exercise 21E.2 you may be given a binomial scenario – for example, a rare defect in a large batch – and asked to approximate it using the Poisson model to simplify calculations.

当 n 很大而 p 很小时，二项分布 B(n, p) 可以用泊松分布近似，其中 λ = np。通常当 n > 50 且 np < 5，或 n > 20 且 p < 0.1 时，认为近似效果良好。在练习 21E.2 中，你可能会遇到二项分布的场景——例如，大批量生产中的稀有缺陷——然后要求用泊松模型进行近似以简化计算。

For instance, if a machine produces bolts with a 0.4% defect rate and a sample of 600 is taken, the number of defective bolts X ~ B(600, 0.004) can be approximated by X ~ Po(2.4). Then P(X = 3) ≈ (2.4³ × e⁻²·⁴) / 3! ≈ 0.209.

例如，如果一台机器生产螺栓的次品率为 0.4%，从中抽取 600 个样本，则次品数量 X ~ B(600, 0.004) 可近似为 X ~ Po(2.4)。那么 P(X = 3) ≈ (2.4³ × e⁻²·⁴) / 3! ≈ 0.209。

9. Using Statistical Tables | 使用泊松分布表

In some IB questions or older past papers, you might be given an extract from Poisson cumulative probability tables. These tables list P(X ≤ x) for various values of λ. To read them, locate the column for your λ and the row for the desired x. The intersection gives P(X ≤ x). From that you can derive other probabilities, just as you would with a calculator.

在一些 IB 考题或较早的历年试卷中，你可能会拿到一张泊松累积概率表的摘录。这些表列出了不同 λ 值下 P(X ≤ x) 的值。要读取表格，找到 λ 所在的列和所需 x 所在的行，交会处的数值就是 P(X ≤ x)。你可以像使用计算器一样，从中推导出其他概率。

Examining table patterns also helps develop an intuition: as λ increases, the distribution spreads out and becomes more symmetric. Exercise 21E.2 might include a table-based question, so it is wise to practise interpreting them rapidly.

观察表格中的规律还有助于培养直觉：随着 λ 增大，分布会变得越来越分散且趋于对称。练习 21E.2 中可能包含基于表格的题目，因此快速解读表格的练习是明智的。

10. Recognising Poisson Scenarios | 识别泊松场景

Typical situations modelled by the Poisson distribution include: the number of phone calls arriving at a switchboard in one hour, the number of accidents at a junction per week, the number of misprints per page in a book, or the number of radioactive decays in a fixed time interval. In Exercise 21E.2, you must read the context carefully to extract λ and identify the time or space interval.

泊松分布建模的典型情景包括：一小时内到达交换机台的电话数量、一周内某路口的交通事故数量、一本书每页的印刷错误数量，或固定时间间隔内的放射性衰变次数。在练习 21E.2 中，你需要仔细阅读上下文，提取出 λ 并识别时间或空间间隔。

If the interval in the question is different from the one for which the rate is given, you must adjust λ proportionally. For instance, if the mean is 8 per hour and you need the probability for a 15‑minute period, then λ for 15 minutes is 8 × (15/60) = 2. This scaling principle is tested repeatedly.

如果题目中的间隔与给出速率的间隔不同，你必须按比例调整 λ。例如，如果平均值为每小时 8 次，而你需要计算 15 分钟内的概率，那么 15 分钟的 λ 为 8 × (15/60) = 2。这一缩放原则经常被考查。

11. Common Mistakes and How to Avoid Them | 常见错误及避免方法

A frequent error is using the wrong tail probability. Students often confuse P(X > a) with P(X ≥ a). Remember that for discrete distributions P(X ≥ a) = 1 – P(X ≤ a – 1). Another pitfall is forgetting to adjust λ when the time interval changes. Double‑check that the λ you use matches the interval required.

一个常见错误是使用了错误的尾部概率。学生经常混淆 P(X > a) 和 P(X ≥ a)。请记住，对于离散分布，P(X ≥ a) = 1 – P(X ≤ a – 1)。另一个陷阱是当时间间隔变化时忘记调整 λ。务必仔细检查你所使用的 λ 是否与所需的时间间隔相匹配。

Also avoid early rounding: keep intermediate λ values and exponential terms in your calculator’s memory to three or four decimal places, then round only the final answer. Lastly, be careful with the words ‘exactly’, ‘at most’ and ‘at least’, and always translate them into a symbolic probability statement before calculating.

此外，要避免过早四舍五入：将中间的 λ 值和指数项保留在计算器存储器中，保留三位或四位小数，仅对最终答案进行四舍五入。最后，注意“恰好”“至多”和“至少”这些词语，在计算前一定要先将它们转换为符号化的概率陈述。

12. Summary and Key Takeaways | 总结与关键要点

Exercise 21E.2 consolidates the core skills for mastering the Poisson distribution. Always define the random variable, state the distribution and its parameter, write the probability statement, and then compute. Use poissonpdf for exact probabilities and poissoncdf for cumulative ones. When combining independent Poisson counts, add the rates; when approximating a binomial, set λ = np and verify the conditions.

练习 21E.2 巩固了掌握泊松分布所需的核心技能。始终定义随机变量，说明分布及其参数，写出概率陈述，然后再进行计算。用 poissonpdf 处理精确概率，用 poissoncdf 处理累积概率。合并独立泊松计数时，将速率相加；近似二项分布时，设 λ = np 并验证条件。

By working through each problem methodically, you will become fluent in switching between algebraic formula, calculator syntax, tables and real‑world language. This fluency is exactly what is expected for top marks in IB Mathematics. As a final tip, re‑attempt any question you got wrong and explain the solution aloud – this deepens retention.

通过有条理地处理每个问题，你将能熟练地在代数公式、计算器语法、表格和现实世界语言之间进行切换。这种熟练度正是 IB 数学高分所要求的。最后一个小建议：重新尝试所有做错的题目，并大声讲解解题过程——这可以加深记忆。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Exercise 21E.2: Poisson Distribution | 练习21E.2：泊松分布