📚 Poisson Distribution | 泊松分布
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. It is widely used in fields such as queueing theory, insurance, telecommunications and natural phenomena. In IB Mathematics Analysis & Approaches and Applications & Interpretation HL, you will learn its fundamentals, how to compute probabilities, and how to apply it to approximate the binomial distribution under certain conditions.
泊松分布是一种离散概率分布，它表示在固定的时间或空间间隔内发生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，并且与上次事件以来的时间无关。它广泛应用于排队论、保险、电信和自然现象等领域。在IB数学分析与方法以及应用与解释的高级课程中，你将学习它的基本原理、如何计算概率，以及如何在特定条件下用它来近似二项分布。
1. What is the Poisson Distribution? | 什么是泊松分布？
The Poisson distribution describes the number of events, X, that occur in a fixed interval. The single parameter, lambda (λ), represents the average number of occurrences in that interval. Typical examples include the number of phone calls received by a call centre per hour, the number of decay events from a radioactive source per second, or the number of typing errors on a page of text.
泊松分布描述了在固定区间内发生的事件次数 X。它的唯一参数 λ（拉姆达）表示在该区间内事件发生的平均次数。典型的例子包括呼叫中心每小时接到的电话数、放射源每秒的衰变事件数，或者一页文本上的打字错误数。
Because the distribution models counts of events, the random variable X can take values 0, 1, 2, 3, … with no theoretical upper bound. Each value has a probability determined by λ. The distribution is skewed for small λ and becomes more symmetric as λ increases.
由于该分布对事件计数进行建模，随机变量 X 可以取 0, 1, 2, 3……等数值，理论上没有上界。每个值都有由 λ 决定的概率。当 λ 较小时，分布呈偏态；随着 λ 增大，分布变得更加对称。
2. Conditions for a Poisson Model | 泊松模型的适用条件
For a Poisson distribution to be an appropriate model, the following conditions should be satisfied:
- Events occur independently of one another.
- The average rate at which events occur is constant throughout the interval.
- Two events cannot occur at exactly the same instant.
- The probability of an event occurring in a very short sub-interval is proportional to the length of that sub-interval.
要使用泊松分布作为合适的模型，应满足以下条件：
- 事件相互独立地发生。
- 事件发生的平均速率在整个间隔内是恒定的。
- 两个事件不可能在同一瞬间发生。
- 在一个极短的子区间内发生事件的概率与该子区间的长度成正比。
In practice, these conditions are often approximated. For example, the number of cars passing a checkpoint per minute can be modelled by a Poisson distribution if traffic is free‑flowing and the rate is roughly constant. However, during traffic jams the independence and constant rate assumptions break down.
在实践中，这些条件通常是近似满足的。例如，如果交通畅通且速率大致恒定，每分钟通过检查站的汽车数量就可以用泊松分布来建模。然而，在交通堵塞期间，独立性和恒定速率的假设就不再成立了。
3. The Probability Mass Function | 概率质量函数
If X ~ Po(λ), then the probability that X takes the value x (where x = 0, 1, 2, …) is given by the probability mass function:
若 X 服从参数为 λ 的泊松分布，即 X ~ Po(λ)，则 X 取值为 x（x = 0, 1, 2, …）的概率由概率质量函数给出：
P(X = x) = (e−λ × λ^x) / x!
Here e ≈ 2.71828 is Euler's number, λ is the mean number of events per interval, and x! denotes x factorial. The expression is defined for all non‑negative integers x, and it sums to 1 over all possible x.
这里 e ≈ 2.71828 是自然对数的底数，λ 是每个间隔的平均事件数，x! 表示 x 的阶乘。该表达式对所有非负整数 x 都有定义，并且所有可能的 x 对应的概率之和为 1。
ibution is determined. You can show that the sum of the probabilities converges to e^λ × e−λ = 1, confirming it is a valid probability distribution.
因为 λ 是唯一的参数，一旦 λ 已知，整个分布就确定了。你可以证明这些概率的和等于 e^λ × e−λ = 1，从而确认它是一个有效的概率分布。
4. Mean and Variance | 均值与方差
One of the remarkable properties of the Poisson distribution is that its mean and variance are both equal to λ:
E(X) = λ, Var(X) = λ
泊松分布的一个显著性质是其均值与方差都等于 λ：
E(X) = λ， Var(X) = λ
√λ. Because the variance equals the mean, the Poisson distribution is often called an equi‑ model may be inadequate, and a negative binomial distribution might be considered instead.
这意味着标准差为 √λ。由于方差等于均值，泊松分布常被称为等离散分布。在实际数据中，如果样本方差远大于样本均值，泊松模型可能不合适，此时可考虑使用负二项分布。
ion. For IB examinations you are expected to know and use E(X) = Var(X) = λ without proof.
E(X) = λ 的结果是直观的：λ 就是平均发生次数。方差与之相等则源于 PMF 的特殊形式，可以用矩母函数或直接求和来证明。在 IB 考试中，你需要记住并运用 E(X) = Var(X) = λ，但无需给出证明。
5. Calculating Individual Probabilities | 计算个别概率
When λ is small, manual calculation using the PMF is manageable. For example, suppose X ~ Po(2). Find P(X = 3).
当 λ 较小时，使用 PMF 手动计算是可以的。例如，假设 X ~ Po(2)，求 P(X = 3)。
Using the formula: P(X = 3) = (e−2 × 23) / 3! = (0.1353… × 8) / 6 ≈ 0.1804. So the probability of exactly three events is about 0.180.
使用公式：P(X = 3) = (e−2 × 2³) / 3! = (0.1353… × 8) / 6 ≈ 0.1804。因此恰好发生三次事件的概率约为 0.180。
For larger λ or multiple calculations, a graphic display calculator (GDC) is typically used. In IB exams you are allowed to use the Poisson probability density function (poissonpdf) and cumulative distribution function (poissoncdf) on your GDC. It is important to interpret the question correctly: 'exactly' requires pdf, while 'at most' or 'more than' needs cdf.
对于较大的 λ 或多次计算，通常使用图形计算器。在 IB 考试中，你可以在计算器上使用泊松概率密度函数（poissonpdf）和累积分布函数（poissoncdf）。正确理解题意非常重要：“恰好”需要 pdf，而“至多”或“超过”则需要 cdf。
6. Using Cumulative Poisson Probabilities | 使用累积泊松概率
Poisson distribution is discrete, careful attention must be paid to the inequality signs.
累积概率用于处理诸如 P(X ≤ k)、P(X < k)、P(X ≥ k) 和 P(X > k) 的问题。由于泊松分布是离散的，务必注意不等号的方向。
iable, so use P(X ≤ 3). Similarly, P(X ≥ 5) = 1 − P(X ≤ 4) and P(X > 5) = 1 − P(X ≤ 5).
例如，P(X ≤ 4) 可以直接用 cdf 函数求得。要计算 P(X < 4)，应注意到对于离散计数变量，X < 4 等价于 X ≤ 3，因此使用 P(X ≤ 3)。类似地，P(X ≥ 5) = 1 − P(X ≤ 4)，而 P(X > 5) = 1 − P(X ≤ 5)。
ers values 0 and 1. Always draw a number line or list integer values if you are uncertain.
考试题常要求计算某个区间的概率，如 P(2 ≤ X ≤ 7)。这可以计算为 P(X ≤ 7) − P(X ≤ 1)，因为 P(X ≤ 1) 包含了 0 和 1。如果不确定，可以画一条数轴或列出整数值。
7. Poisson Approximation to the Binomial | 泊松分布对二项分布的近似
When the number of trials n in a binomial distribution is large and the probability of success p is small, such that the product np remains moderate, the Poisson distribution provides an excellent approximation. The rule of thumb often quoted is n > 50 and np < 5, or more generally np < 10.
当二项分布中的试验次数 n 很大、成功概率 p 很小，且乘积 np 保持适中时，泊松分布可以给出极佳的近似。常用的经验法则是 n > 50 且 np < 5，或者更一般地 np < 10。
If X ~ This simplifies calculations because the Poisson PMF involves only one parameter instead of two, and avoids computing large binomial coefficients.
如果 X ~ B(n, p)，且 n 很大、p 很小，那么 X 近似服从参数 λ = np 的泊松分布。这简化了计算，因为泊松 PMF 只涉及一个参数，并且无需计算巨大的二项式系数。
s B(1000, 0.002). Using the Poisson approximation, λ = 1000 × 0.002 = 2. Then P(X = 0) ≈ e−2 × 20 / 0! ≈ 0.1353, which is very close to the exact binomial probability of 0.1351.
例如，假设某零件有缺陷的概率为 0.002，一批产品包含 1000 个零件。缺陷数量的确切分布为 B(1000, 0.002)。使用泊松近似，λ = 1000 × 0.002 = 2。于是 P(X = 0) ≈ e−2 × 2⁰ / 0! ≈ 0.1353，这与精确的二项概率 0.1351 非常接近。
8. 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 also follows a Poisson distribution:
X + Y ~ Po(λ₁ + λ₂)
如果 X ~ Po(λ₁) 且 Y ~ Po(λ₂) 是独立的随机变量，那么它们的和也服从泊松分布：
X + Y ~ Po(λ₁ + λ₂)
This additive property extends to any finite number of independent Poisson variables: the sum of independent Po(λᵢ) variables is Po(Σ λᵢ). It is particularly useful when merging independent event streams, such as combining the arrivals at two separate ticket counters into a total arrivals process.
这一可加性可以推广到任意有限个独立的泊松变量：多个独立的 Po(λᵢ) 变量之和服从 Po(Σ λᵢ)。这在合并独立事件流时特别有用，例如将两个独立售票处的到达人数合并为总的到达过程。
A common IB HL question gives independent Poisson variables and asks for the distribution of their total, or the probability that the total exceeds a certain value. You must remember to check the independence condition and then simply add the parameters.
IB 高级课程中常见的问题是给出独立的泊松变量，要求求其总和的分布，或者总和超过某值的概率。你必须记得核查独立性条件，然后直接把参数相加即可。
9. Common Mistakes and Exam Tips | 常见错误与考试技巧
One frequent mistake is confusing the Poisson distribution with the binomial. Always ask: is there a fixed number of trials? If the context counts events in a continuous interval without an upper bound, the Poisson is usually appropriate. If there is a fixed number of trials n and a constant probability p, use the binomial.
一个常见的错误是将泊松分布与二项分布混淆。务必自问：是否存在固定的试验次数？如果情境是在连续区间内计数事件，且没有上限，则通常适合用泊松分布。如果存在固定的试验次数 n 和恒定的成功概率 p，则应使用二项分布。
Another typical error occurs with cumulative inequalities. Always convert 'less than' to 'less than or equal to one less' ibutions: P(X < a) = P(X ≤ a − 1) for integer a. Similarly, P(X > a) = 1 − P(X ≤ a).
另一个典型错误出现在累积不等式上。对于离散分布，务必把“小于”转化为“小于或等于少一”：对于整数 a，有 P(X < a) = P(X ≤ a − 1)。类似地，P(X > a) = 1 − P(X ≤ a)。
andard deviation is √λ. If a question asks for the standard deviation, do not just write λ – take the square root.
考生有时会误用均值和方差。记住，对于泊松分布，方差等于均值，因此标准差为 √λ。如果题目要求标准差，不要只写 λ——需要开平方根。
n approximation to the binomial, check that the conditions are satisfied. A statement like 'the number of trials is large and the probability of success is small' is usually sufficient, but numerical justification (e.g. n = 200, p = 0.01 ⇒ np = 2) strengthens your argument in an exam.
在使用泊松分布近似二项分布时，要验证条件是否满足。“试验次数很大且成功概率很小”这样的陈述通常就足够了，但如果能给出数值依据（例如 n = 200, p = 0.01 ⇒ np = 2），在考试中更能增强你的论证。
Finally, always show your working, even if the GDC gives the answer directly. Write down the distribution (e.g. X ~ Po(3)) and the probability statement you are evaluating before citing the calculator result. This gains method marks and helps avoid careless errors.
最后，即使图形计算器直接给出了答案，也务必展示你的步骤。写出分布（如 X ~ Po(3)）以及你正在求值的概率表达式，然后再引用计算器结果。这能获得方法分，并有助于避免粗心错误。
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