📚 Poisson Distributions | 泊松分布
The Poisson distribution is a discrete probability distribution that models the number of times an event occurs in a fixed interval of time or space. It is named after the French mathematician Siméon Denis Poisson and appears throughout the Edexcel A-Level Statistics course.
泊松分布是一种离散概率分布，用于模拟在固定的时间或空间区间内某事件发生的次数。它以法国数学家西梅翁·德尼·泊松命名，贯穿于爱德思 A-Level 统计学课程。
1. What is a Poisson Distribution? | 什么是泊松分布？
A random variable X has a Poisson distribution with parameter λ if it counts the number of events occurring in a fixed interval when events happen at a constant average rate. We write X ~ Po(λ). The parameter λ can be any positive real number, and X can take any non-negative integer value: 0, 1, 2, 3, ... . Typical examples include the number of calls received by a call centre in an hour, the number of flaws per metre of fabric, or the number of cars passing a checkpoint in a minute.
随机变量 X 若在固定区间内以恒定的平均速率计数事件发生的次数，则服从参数为 λ 的泊松分布，记作 X ~ Po(λ)。参数 λ 可以是任意正实数，X 取任意非负整数值：0、1、2、3……。典型例子包括呼叫中心一小时内接到的电话数、每米织物上的瑕疵数、或一分钟内通过检查点的汽车数。
2. Conditions for a Poisson Model | 泊松模型的条件
A Poisson model is appropriate only when three main conditions hold. First, events must occur independently: knowing that one event has occurred should not change the probability of another event in the same interval. Second, events must occur singly, meaning two events cannot happen at exactly the same instant. Third, the average rate of occurrence, λ, must be constant throughout the interval. In exam questions, these conditions are often stated in context, such as 'calls arrive at a constant rate and independently'.
只有当三个主要条件满足时，泊松模型才适用。第一，事件必须独立发生：已知一个事件已发生，不应改变同一区间内另一事件发生的概率。第二，事件必须单个发生，即两个事件不能在同一瞬间发生。第三，平均发生率 λ 必须在整个区间内保持恒定。在考试题中，这些条件通常会在情境中给出，例如“电话以恒定速率独立到达”。
3. The Poisson Probability Formula | 泊松概率公式
If X ~ Po(λ), then the probability that X takes a particular value x is given by the formula:
如果 X ~ Po(λ)，则 X 取某个特定值 x 的概率由以下公式给出：
P(X = x) = (λˣ e⁻λ) / x! , x = 0, 1, 2, ...
Here e is Euler's number, approximately 2.71828, and x! denotes x factorial, with 0! = 1. For example, if λ = 2, then P(X = 3) = (2³ e⁻²) / 3! = 8e⁻² / 6 ≈ 0.1804.
其中 e 是欧拉数，约等于 2.71828；x! 表示 x 的阶乘，且 0! = 1。例如，若 λ = 2，则 P(X = 3) = (2³ e⁻²) / 3! = 8e⁻² / 6 ≈ 0.1804。
4. Mean and Variance | 均值与方差
A key property of the Poisson distribution is that its mean and variance are both equal to λ. That is:
泊松分布的一个关键性质是其均值与方差都等于 λ。即：
E(X) = λ, Var(X) = λ
ck whether a Poisson model is reasonable: if a sample has a mean much smaller or larger than its variance, a different distribution may be needed.
因此标准差为 √λ。这一等式常用于检验泊松模型是否合理：如果样本的均值远小于或远大于方差，则可能需要使用其他分布。
| Expectation / 期望 | E(X) = λ |
| Variance / 方差 | Var(X) = λ |
| Standard deviation / 标准差 | √λ |
5. Using Cumulative Poisson Probabilities | 使用泊松累积概率
≤ k). These can be found from statistical tables or a calculator. Remember the identities:
考试题经常要求计算累积概率，如 P(X ≤ k)。这些概率可以通过统计表或计算器得到。请记住以下恒等式：
P(X > k) = 1 − P(X ≤ k)
P(X ≥ k) = 1 − P(X ≤ k−1)
P(X = k) = P(X ≤ k) − P(X ≤ k−1)
For example, if X ~ Po(3), then P(X ≤ 2) = 0.4232 from tables.
例如，若 X ~ Po(
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