Poisson Distribution: Essential Points for IB Maths | 泊松分布：IB数学要点解析

📚 Poisson Distribution: Essential Points for IB Maths | 泊松分布：IB数学要点解析

The Poisson distribution is one of the most elegant and practical probability models in the IB Mathematics curriculum. It models the number of events occurring in a fixed interval of time or space, making it indispensable for fields ranging from biology to telecommunications. This guide consolidates every essential point you need for your IB exams.

泊松分布是 IB 数学课程中最优雅且最实用的概率模型之一。它用于描述在固定时间或空间区间内事件发生的次数，在生物学到电信等众多领域中都不可或缺。本篇指南为你整合了 IB 考试所需的全部要点。

1. Definition and Conditions | 定义与条件

A random variable X follows a Poisson distribution if it counts the number of events occurring in a fixed interval, provided the following conditions hold:

若随机变量 X 用于统计固定区间内事件发生的次数，且满足以下条件，则 X 服从泊松分布：

Independence / 独立性: Events occur independently of one another. | 事件之间相互独立，互不影响。
Single occurrence / 单一发生: Two or more events cannot occur at exactly the same instant. | 两个或两个以上事件不能恰好同时发生。
Constant rate / 恒定速率: The average rate λ (lambda) is constant throughout the interval. | 平均发生率 λ（拉姆达）在整个区间内保持恒定。

X ~ Po(λ)

Here, λ represents the mean number of occurrences in the given interval. Note that λ is both the mean AND the variance of the distribution — a signature property.

这里的 λ 表示给定区间内事件的平均发生次数。请注意，λ 既是分布的均值也是方差——这是泊松分布的一个标志性性质。

2. Probability Mass Function | 概率质量函数

If X ~ Po(λ), the probability of observing exactly x events is given by the probability mass function (PMF):

若 X ~ Po(λ)，恰好观察到 x 个事件的概率由概率质量函数（PMF）给出：

P(X = x) = (e⁻λ × λˣ) / x! , where x = 0, 1, 2, …

e is Euler’s number (approximately 2.71828) | e 是欧拉数（约等于 2.71828）
x! denotes the factorial of x | x! 表示 x 的阶乘
x can take any non-negative integer value | x 可以取任何非负整数值

The IB formula booklet provides this formula, but you must know how to apply it correctly, especially in calculator-based questions where you will use the Poisson PD function.

IB 公式手册中提供了此公式，但你必须知道如何正确运用它，尤其是在需要使用泊松概率密度函数（Poisson PD）求解的计算器题目中。

3. Mean and Variance | 均值与方差

A defining characteristic of the Poisson distribution is that its mean and variance are equal.

泊松分布的一个决定性特征是它的均值与方差相等。

E(X) = λ and Var(X) = λ

Equivalently, the standard deviation is √λ. This equality is unique among common distributions and is often used to test whether data might follow a Poisson model.

等价地，标准差为 √λ。这种相等关系在常见分布中是独一无二的，通常用于检验数据是否符合泊松模型。

When solving problems, you may need to find λ from given information. For instance, if the mean is given as 5, then both E(X) and Var(X) equal 5. If only the variance is provided, λ equals that variance.

解题时，你可能需要根据已知信息求出 λ。例如，若均值为 5，则 E(X) 和 Var(X) 都等于 5。如果只给出方差，那么 λ 就等于该方差。

4. Changing the Interval | 区间变换

A critical skill is scaling λ when the interval changes. If X ~ Po(λ) for a unit interval, then for an interval of length t, the new rate is λt.

一个关键技能是在区间变化时对 λ 进行缩放。如果 X ~ Po(λ) 对应单位区间，那么对于长度为 t 的区间，新的速率是 λt。

Y ~ Po(λt)

Example / 示例: If the number of cars passing a point follows Po(4) per minute, then in 5 minutes, the number of cars follows Po(20).

如果每分钟经过某点的汽车数服从 Po(4)，那么 5 分钟内经过的汽车数服从 Po(20)。

This principle is essential for word problems that ask for probabilities over different time spans, areas, or volumes. Always convert λ to match the interval in the question.

这一原理对于涉及不同时间段、面积或体积概率的文字题至关重要。务必根据题目中的区间调整 λ 的取值。

5. Addition of Independent Poisson Variables | 独立泊松变量的加法

If X and Y are independent Poisson random variables such that X ~ Po(λ₁) and Y ~ Po(λ₂), then their sum also follows a Poisson distribution:

如果 X 和 Y 是独立的泊松随机变量，且 X ~ Po(λ₁)、Y ~ Po(λ₂)，那么它们的和同样服从泊松分布：

X + Y ~ Po(λ₁ + λ₂)

This additive property is powerful when dealing with combined sources of events, such as total calls received on two phone lines.

这一可加性质在处理多来源事件时非常强大，例如两条电话线收到的总呼叫数量。

Example / 示例: X ~ Po(3) and Y ~ Po(4), independent. Then X + Y ~ Po(7). To find P(X + Y = 5), simply use Po(7).

设 X ~ Po(3)，Y ~ Po(4)，且相互独立。则 X + Y ~ Po(7)。要求 P(X + Y = 5)，直接使用 Po(7) 即可。

6. Approximation to the Binomial Distribution | 二项分布的泊松近似

The Poisson distribution can approximate the binomial distribution when n is large and p is small. The rule of thumb you must know:

当 n 很大且 p 很小时，泊松分布可以近似二项分布。你必须记住的经验法则：

If n ≥ 50 and p ≤ 0.1, then B(n, p) ≈ Po(np)

In practice, this approximation works extremely well when n ≥ 50 and p ≤ 0.1. Some textbooks require n ≥ 20 and p ≤ 0.05; the IB commonly accepts the condition n is large and p is small.

在实践中，当 n ≥ 50 且 p ≤ 0.1 时，该近似效果极佳。有些教材要求 n ≥ 20 且 p ≤ 0.05；IB 通常接受的条件是 n 足够大且 p 足够小。

The approximation relies on the fact that for small p, the binomial probability P(X = x) = C(n, x)pˣ(1−p)ⁿ⁻ˣ converges to the Poisson form (e⁻ᵑᵖ(np)ˣ)/x!.

该近似基于以下事实：当 p 很小时，二项概率 P(X = x) = C(n, x)pˣ(1−p)ⁿ⁻ˣ 收敛于泊松形式 (e⁻ᵑᵖ(np)ˣ)/x!。

7. Using Your GDC | 图形计算器的使用

Proficiency with your GDC is essential for IB exams. Here are the key functions you must locate:

熟练掌握图形计算器（GDC）对 IB 考试至关重要。以下是必须找到的关键功能：

Poisson PD: Computes P(X = x) directly | 直接计算 P(X = x)
Poisson CD: Computes P(X ≤ x) or P(X ≥ x) cumulatively | 累加计算 P(X ≤ x) 或 P(X ≥ x)
List-based data: You may be asked to generate a probability table | 列表演算：你可能会被要求生成概率表

In the exam, for P(X ≥ k), it is often easier to compute 1 − P(X ≤ k − 1), since most calculators provide upper-tail probabilities directly.

在考试中，计算 P(X ≥ k) 时，通常更简便的方法是计算 1 − P(X ≤ k − 1)，因为大多数计算器直接提供上尾概率。

Calculator tip / 计算器技巧: Write down the GDC functions you use (e.g., “Poisson CD λ = 4, x = 3”) as evidence of your working method.

写下你使用的计算器功能（例如 “Poisson CD λ = 4, x = 3″）作为解题过程的证明。

8. Typical Exam Question Types | 典型考题类型

IB examiners love to test Poisson distribution through word problems and multi-part questions. Common scenarios include:

IB 考官喜欢通过应用题和多部分问题来考查泊松分布。常见题型包括：

Direct probability / 直接概率: Calculate P(X = x) or P(X ≤ x) given λ | 已知 λ，计算 P(X = x) 或 P(X ≤ x)
Interval scaling / 区间缩放: Finding probabilities over a different time or area | 求不同时间或面积下的概率
Conditional probability / 条件概率: Using the formula P(A|B) = P(A∩B)/P(B) | 使用公式 P(A|B) = P(A∩B)/P(B)
Real-world modelling / 现实建模: Judging whether a Poisson model is suitable (checking conditions) | 判断泊松模型是否合适（检验条件）

A common trap is forgetting to multiply λ when the interval changes — always read the question carefully to determine the unit interval.

一个常见陷阱是区间变化时忘记缩放 λ——务必仔细阅读题目，确定单位区间是什么。

9. Worked Example | 完整示例

Question / 题目: A hospital receives, on average, 3 emergency calls per hour. The number of calls follows a Poisson distribution. Find the probability that (a) exactly 2 calls are received in one hour; (b) at least 4 calls are received in 2 hours.

某医院平均每小时接到 3 个急救电话。来电数服从泊松分布。求：(a) 一小时内恰好接到 2 个电话的概率；(b) 两小时内至少接到 4 个电话的概率。

Solution (a) / 解答 (a):

X ~ Po(3), P(X = 2) = (e⁻³ × 3²) / 2! = (e⁻³ × 9) / 2 ≈ 0.224

Solution (b) / 解答 (b):

For 2 hours, λ becomes 3 × 2 = 6. Thus Y ~ Po(6).

对于 2 小时，λ 变为 3 × 2 = 6。因此 Y ~ Po(6)。

P(Y ≥ 4) = 1 − P(Y ≤ 3)

Using the Poisson CD function with λ = 6 and x = 3, we obtain P(Y ≤ 3) ≈ 0.151. Therefore P(Y ≥ 4) ≈ 1 − 0.151 = 0.849.

使用 Poisson CD 功能，输入 λ = 6 和 x = 3，得到 P(Y ≤ 3) ≈ 0.151。因此 P(Y ≥ 4) ≈ 1 − 0.151 = 0.849。

10. Common Mistakes and How to Avoid Them | 常见错误与规避方法

Even high-achieving students slip up on the following points. Memorise these warnings:

即使是高分学生也会在这几点上出错。请牢记以下提醒：

Mistake 1 / 错误 1: Using λ instead of λt after an interval change. | 区间变化后仍然使用 λ 而非 λt。
Mistake 2 / 错误 2: Confusing P(X ≥ x) with 1 − P(X ≤ x). The correct complement is 1 − P(X ≤ x − 1). | 混淆 P(X ≥ x) 与 1 − P(X ≤ x)。正确的补是 1 − P(X ≤ x − 1)。
Mistake 3 / 错误 3: Applying the binomial approximation without checking n ≥ 50 and p ≤ 0.1. | 未检查 n ≥ 50 且 p ≤ 0.1 就使用二项近似。
Mistake 4 / 错误 4: Forgetting that for Poisson, mean = variance = λ. | 忘记泊松分布中均值 = 方差 = λ。
Mistake 5 / 错误 5: Not specifying the distribution in your working, losing method marks. | 解题时未明确写出分布，导致丢失方法分。

11. Relationship with the Normal Distribution | 与正态分布的关系

For large λ (typically λ ≥ 15), the Poisson distribution can be approximated by a normal distribution with mean λ and variance λ.

当 λ 较大时（通常 λ ≥ 15），泊松分布可以用正态分布来近似，该正态分布的均值为 λ，方差也为 λ。

Po(λ) ≈ N(λ, λ) for λ ≥ 15

When applying this approximation, you must use a continuity correction. For example, P(X ≤ x) becomes P(Y ≤ x + 0.5). This improves the accuracy of the approximation.

使用此近似时，必须进行连续性修正。例如，P(X ≤ x) 变为 P(Y ≤ x + 0.5)。这会提高近似的准确性。

Continuity corrections follow the same rules as those used for the binomial-to-normal approximation: subtract 0.5 for ≥, add 0.5 for ≤.

连续性修正的规则与二项分布到正态分布近似相同：求 ≥ 时减去 0.5，求 ≤ 时加上 0.5。

12. Summary of Essential Formulas | 必备公式总结

Consolidate your revision with this quick-reference table:

用这个速查表来巩固你的复习：

Property / 性质	Formula / 公式
Probability mass function / 概率质量函数	P(X = x) = (e⁻λ × λˣ) / x!
Mean / 均值	E(X) = λ
Variance / 方差	Var(X) = λ
Interval scaling / 区间缩放	Po(λ) over t units → Po(λt)
Addition rule / 加法规则	Po(λ₁) + Po(λ₂) = Po(λ₁ + λ₂)
Binomial approximation / 二项近似	B(n, p) ≈ Po(np) if n ≥ 50, p ≤ 0.1
Normal approximation / 正态近似	Po(λ) ≈ N(λ, λ) if λ ≥ 15

Keep this table close during your revision sessions, and always write the distribution notation X ~ Po(λ) in your exam working to demonstrate your understanding.

复习时请随身携带此表，并在考试答题时写出分布记号 X ~ Po(λ)，以展示你的理解。

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Poisson Distribution: Essential Points for IB Maths | 泊松分布：IB数学要点解析