📚 Poisson Cumulative Distribution Function | 泊松累积分布函数
The Poisson cumulative distribution function (CDF) is one of the most frequently examined topics in Edexcel A-Level Further Mathematics. It allows us to calculate the probability that a Poisson random variable takes a value less than or equal to a given number, and it forms the foundation for solving nearly all exam questions on the Poisson distribution.
泊松累积分布函数（CDF）是爱德思（Edexcel）A-Level 进阶数学中考试频率最高的考点之一。它使我们能够计算泊松随机变量取小于或等于某个给定值的概率，也是解答几乎所有泊松分布考题的基础。
1. What is the Poisson Distribution? | 什么是泊松分布
A Poisson distribution is used to model the number of times an event occurs in a fixed interval of time or space, provided that the events occur independently and at a constant average rate. Common examples include the number of phone calls received in an hour, the number of accidents on a road per day, or the number of bacteria in a fixed volume of liquid.
泊松分布用于对固定时间或空间区间内事件发生的次数进行建模，前提是事件独立发生且平均发生率恒定。常见例子包括一小时内接到的电话数、一条道路每天的事故数，或固定体积液体中的细菌数量。
We write X ~ Po(λ), where λ (lambda) represents the mean number of events in that interval. The distribution is discrete, so X can only take non-negative integer values 0, 1, 2, …
我们记作 X ~ Po(λ)，其中 λ（lambda）表示该区间内事件的平均发生次数。该分布是离散型的，因此 X 只能取非负整数 0, 1, 2, …
2. The Probability Mass Function | 概率质量函数
The probability that X takes exactly the value r is given by the probability mass function (PMF):
随机变量 X 恰好取值为 r 的概率由概率质量函数（PMF）给出：
P(X = r) = e−λ × λr / r! , for r = 0, 1, 2, …
Here, e ≈ 2.718 is the base of the natural logarithm, λ is the mean of the distribution, and r! denotes r factorial. For example, if X ~ Po(2.5), then P(X = 0) = e−2.5 × 2.50 / 0! = e−2.5 ≈ 0.0821.
其中 e ≈ 2.718 是自然对数的底数，λ 是分布的均值，r! 表示 r 的阶乘。例如，若 X ~ Po(2.5)，则 P(X = 0) = e−2.5 × 2.50 / 0! = e−2.5 ≈ 0.0821。
Every individual probability is positive, and summing over all possible values of r gives exactly 1. This property is built into the formula, so no normalisation is needed.
每一个概率都是正的，并且对所有可能的 r 求和恰好等于 1。这一性质内建于公式之中，因此无需再做归一化处理。
3. The Cumulative Distribution Function | 累积分布函数
The cumulative distribution function F(x) = P(X ≤ x) is defined as the sum of all individual probabilities from X = 0 up to X = x:
累积分布函数 F(x) = P(X ≤ x) 定义为从 X = 0 到 X = x 的所有单个概率之和：
P(X ≤ x) = Σr=0x e−λ × λr / r!
Using the formula directly is tedious when x is large. Fortunately, the Edexcel formula booklet provides cumulative probability tables for many commonly used values of λ, and you are expected to read these tables fluently in your exam.
当 x 较大时，直接使用公式求和十分繁琐。幸运的是，爱德思公式书提供了许多常用 λ 值所对应的累积概率表，考试中你需要能够熟练地查读这些表格。
The cumulative function is non-decreasing: as x increases, P(X ≤ x) never decreases, and it approaches 1 as x becomes very large.
累积函数是非递减的：随着 x 增大，P(X ≤ x) 不会减少，并且当 x 非常大时趋近于 1。
4. Reading Cumulative Probability Tables | 查读累积概率表
In the Edexcel formula booklet, the Poisson tables are arranged with λ along one axis and x along the other. Each entry directly gives the value of P(X ≤ x) for that particular λ.
在爱德思公式书中，泊松分布表以 λ 作为一个轴、x 作为另一个轴进行排列，每个表项直接给出该 λ 下 P(X ≤ x) 的数值。
Almost every exam question can be solved by rewriting the required probability in terms of P(X ≤ x). The three most important transformations are shown below:
几乎所有考试题都可以通过将所求概率改写为 P(X ≤ x) 的形式来解答。以下三种变换最为重要：
- P(X = a) = P(X ≤ a) − P(X ≤ a − 1)
- P(X > a) = 1 − P(X ≤ a)
- P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1)
- P(X = a) = P(X ≤ a) − P(X ≤ a − 1)
- P(X > a) = 1 − P(X ≤ a)
- P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1)
Students often forget the “minus 1” in the first and third rules. Remember that the table is cumulative from zero, so to isolate a single value or a closed interval you must subtract everything up to the previous integer.
同学们经常忘记第一条和第三条规则中的“减 1”。请记住表格是从零开始累积的，因此要分离出单一数值或闭区间，就必须减去直到前一个整数为止的所有概率。
5. Mean and Variance | 均值与方差
For a Poisson distribution, the mean and the variance are both equal to λ. This is a unique and very useful property:
对于泊松分布，均值与方差均等于 λ。这是一个独特且非常有用的性质：
E(X) = λ , Var(X) = λ
Since the variance equals λ, the standard deviation is √λ. This means a typical range of observed values is roughly λ ± √λ. For example, if X ~ Po(9), the standard deviation is 3, and most observations would lie between about 6 and 12.
由于方差等于 λ，标准差为 √λ。这意味着观测值的典型范围大约为 λ ± √λ。例如，若 X ~ Po(9)，标准差为 3，绝大多数观测值将落在 6 到 12 之间。
If you are given a sample of data in an exam and asked whether a Poisson model is appropriate, one quick check is whether the sample mean is close to the sample variance.
如果考试中给你一组样本数据，并要求判断泊松模型是否合适，一个快速检验方法就是看样本均值是否接近样本方差。
6. Conditions for Using a Poisson Model | 泊松模型的适用条件
A Poisson distribution is only a valid model when specific conditions are satisfied. The standard conditions are listed below:
泊松分布只有满足特定条件时才是有效的模型。标准条件如下：
- Events occur independently of one another.
- Events occur singly, at a constant average rate.
- Two events cannot occur at exactly the same instant.
- The interval being considered is fixed.
- 事件彼此独立地发生。
- 事件以恒定平均速率逐个发生。
- 两个事件不能在完全同一瞬间发生。
- 所考虑的区间是固定的。
If the average rate is λ per unit interval, then over t units of the interval the distribution becomes Po(λt). For example, if the mean is 3 per hour, then over 2.5 hours the mean becomes 3 × 2.5 = 7.5.
若平均速率为每个单位区间 λ 次，则在 t 个单位区间内分布变为 Po(λt)。例如，若均值为每小时 3 次，则在 2.5 小时内均值变为 3 × 2.5 = 7.5。
7. Poisson as an Approximation to the Binomial | 泊松分布对二项分布的近似
When n is large and p is small, the binomial distribution B(n, p) can be approximated by a Poisson distribution with mean λ = np. This is an essential technique for Edexcel Further Mathematics, because binomial cumulative probabilities for very large n are not printed in the formula booklet.
当 n 很大、p 很小时，二项分布 B(n, p) 可以用均值为 λ = np 的泊松分布来近似。这是爱德思进阶数学中的一项关键技巧，因为公式书不提供 n 很大时的二项累积概率。
A common rule of thumb is that the approximation is good when n ≥ 50 and np ≤ 5. The table below compares an exact binomial calculation with the Poisson approximation for n = 80, p = 0.02.
常用的经验法则是当 n ≥ 50 且 np ≤ 5 时近似效果良好。下表比较了 n = 80、p = 0.02 时精确二项计算与泊松近似的差异。
| x | P(X ≤ x) using B(80, 0.02) | P(X ≤ x) using Po(1.6) |
| 0 | 0.1986 | 0.2019 |
| 1 | 0.5245 | 0.5249 |
| 2 | 0.7843 | 0.7833 |
The values are extremely close, confirming that the Poisson approximation is reliable under the stated conditions.
两者数值非常接近，证实了在上述条件下泊松近似是可靠的。
8. Worked Example 1 — Using the Table Directly | 例题一：直接查表
Suppose X ~ Po(4). Find P(X ≤ 3), P(X = 3) and P(X > 5).
设 X ~ Po(4)，求 P(X ≤ 3)、P(X = 3) 和 P(X > 5)。
From the Edexcel tables with λ = 4, we read the cumulative values directly:
从 λ = 4 的爱德思表中直接读出累积值：
P(X ≤ 3) = 0.4335
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