📚 Mean and Variance of a Poisson Distribution | 泊松分布的均值与方差
The Poisson distribution is one of the most useful discrete probability models in A Level Statistics. In this article we focus on two key summary measures: the mean and the variance, both of which are equal to the parameter λ.
泊松分布是 A Level 统计学中最有用的离散概率模型之一。本文重点介绍两个关键的汇总量：均值与方差，二者都等于参数 λ。
1. What Is a Poisson Distribution? | 什么是泊松分布
A Poisson distribution models the number of times an event occurs in a fixed interval of time or space. It applies when events happen independently, singly, and at a constant average rate. In Edexcel A Level Mathematics, we write X ~ Po(λ), where λ is the average number of occurrences.
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。它适用于事件独立发生、一次只发生一个且平均发生率恒定的情况。在 Edexcel A Level 数学中，我们写作 X ~ Po(λ)，其中 λ 是平均发生次数。
- Events occur one at a time — 事件一次发生一个。
- Events are independent — 事件相互独立。
- The average rate is constant — 平均发生率恒定。
2. The Poisson Probability Formula | 泊松概率公式
If X ~ Po(λ), the probability that X takes a particular non-negative integer value x is given by the formula below. Here e is Euler’s number, approximately 2.71828, and x! means x factorial.
如果 X ~ Po(λ)，则 X 取某个非负整数值 x 的概率由以下公式给出。其中 e 是欧拉数，约等于 2.71828，x! 表示 x 的阶乘。
P(X = x) = e⁻λ λˣ / x! , x = 0, 1, 2, …
This formula is given in the Edexcel formula booklet, but you must be able to use it accurately with your calculator.
该公式在 Edexcel 公式手册中给出，但你必须能够用计算器准确地使用它。
3. Mean of a Poisson Distribution | 泊松分布的均值
For X ~ Po(λ), the mean, or expected value, is E(X) = λ. This is often the first value you should write down in an exam question.
对于 X ~ Po(λ)，均值或期望值为 E(X) = λ。这通常是考试题中你应该首先写下的值。
One way to derive this result is to use the definition E(X) = Σ x P(X = x), summing from x = 0 to infinity. The x = 0 term is zero, so after factoring λ and re-indexing the sum, the remaining series sums to eλ, giving E(X) = λ.
推导该结果的一种方法是使用定义 E(X) = Σ x P(X = x)，从 x = 0 到无穷求和。x = 0 的项为零，因此提取 λ 并对求和重新编号后，剩余级数的和为 eλ，从而得到 E(X) = λ。
E(X) = λ
4. Variance of a Poisson Distribution | 泊松分布的方差
The variance of a Poisson random variable is Var(X) = λ, and therefore the standard deviation is SD(X) = √λ. This is a very useful property because the variance is obtained directly from the same parameter as the mean.
泊松随机变量的方差为 Var(X) = λ，因此标准差为 SD(X) = √λ。这是一个非常有用的性质，因为方差与均值来自同一个参数。
To verify this, we use Var(X) = E(X²) – [E(X)]². For the Poisson distribution, E(X²) = λ² + λ, so Var(X) = λ² + λ – λ² = λ.
为了验证这一点，我们使用 Var(X) = E(X²) – [E(X)]²。对于泊松分布，E(X²) = λ² + λ，所以 Var(X) = λ² + λ – λ² = λ。
Var(X) = λ and SD(X) = √λ
5. Why the Mean Equals the Variance | 为什么均值等于方差
For a Poisson distribution, the mean and the variance are always equal. This is not generally true for other common discrete distributions such as the binomial distribution, where the variance is usually smaller than the mean.
对于泊松分布，均值和方差总是相等。对于其他常见离散分布（如二项分布）通常并非如此，二项分布的方差通常小于其均值。
This property helps you check whether a Poisson model is appropriate. If a data set has a sample mean and sample variance that are very different, a Poisson model is unlikely to fit well.
这一性质有助于你判断泊松模型是否合适。如果一组数据的样本均值与样本方差差异很大，泊松模型不太可能拟合良好。
- Mean = λ and variance = λ — 均值 = λ 且方差 = λ。
- Standard deviation =
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