📚 Testing for the Mean of a Poisson Distribution | 泊松分布均值的假设检验
In A-Level Edexcel Statistics, the Poisson distribution is used to model the number of random events occurring in a fixed interval of time or space. When we want to test a claim about the underlying mean rate λ, we perform a hypothesis test for the mean of a Poisson distribution. This article explains the key methods, critical regions, p-values, and worked examples needed for the exam.
在 A-Level Edexcel 统计学中，泊松分布用于模拟固定时间或空间间隔内随机事件发生的次数。当我们想检验关于潜在平均发生率 λ 的声明时，就需要对泊松分布的均值进行假设检验。本文讲解考试所需的关键方法、临界区域、p 值以及典型例题。
1. The Poisson Model and Its Mean | 泊松模型及其均值
A Poisson random variable X counts the number of events in a given interval. Its single parameter λ is both the mean and the variance, so E(X) = λ and Var(X) = λ. In hypothesis testing, the null hypothesis always fixes the value of λ under the assumption that the claimed rate is true.
泊松随机变量 X 表示给定间隔内事件发生的次数。它唯一的参数 λ 既是均值也是方差，因此 E(X) = λ 且 Var(X) = λ。在假设检验中，零假设总是固定 λ 的值，假定所声称的发生率是正确的。
The test statistic is the observed count x taken from a Poisson(λ) distribution when H₀ is true. Because λ is a single parameter, a Poisson test is simpler than many other tests, but the discreteness of the distribution means that exact significance levels are rarely achieved.
检验统计量是当 H₀ 成立时，取自 Poisson(λ) 分布的观测计数 x。由于 λ 是单一参数，泊松检验比许多其他检验更简单，但该分布的离散性意味着很难恰好达到显著性水平。
2. Stating Hypotheses for a Poisson Mean | 泊松均值假设的提出
Depending on the wording of the problem, we use a one-tailed or two-tailed alternative hypothesis. For a test of the mean of a Poisson distribution, the null hypothesis typically takes one of the forms: H₀: λ = λ₀, H₀: λ ≤ λ₀, or H₀: λ ≥ λ₀.
根据题目的措辞，我们使用单尾或双尾备择假设。对于泊松分布均值的检验，零假设通常采用以下形式之一：H₀: λ = λ₀、H₀: λ ≤ λ₀ 或 H₀: λ ≥ λ₀。
- If we suspect the rate has increased: H₀: λ = λ₀, H₁: λ > λ₀ (upper-tail test).
- If we suspect the rate has decreased: H₀: λ = λ₀, H₁: λ < λ₀ (lower-tail test).
- If we suspect the rate has simply changed: H₀: λ = λ₀, H₁: λ ≠ λ₀ (two-tailed test).
- 如果怀疑发生率增加了：H₀: λ = λ₀，H₁: λ > λ₀（上尾检验）。
- 如果怀疑发生率降低了：H₀: λ = λ₀，H₁: λ < λ₀（下尾检验）。
- 如果怀疑发生率只是发生了变化：H₀: λ = λ₀，H₁: λ ≠ λ₀（双尾检验）。
In Edexcel questions, the null hypothesis may be written as λ = λ₀ even when the alternative is one-sided. Both forms H₀: λ = λ₀ and H₀: λ ≤ λ₀ are accepted for an upper-tail test, but always check the mark scheme for the preferred version.
在 Edexcel 的题目中，即使备择假设是单侧的，零假设也可以写成 λ = λ₀。对于上尾检验，H₀: λ = λ₀ 和 H₀: λ ≤ λ₀ 两种形式都被接受，但一定要查看评分方案中推荐的写法。
3. The Test Statistic and Assumptions | 检验统计量与假设条件
The test statistic is simply the observed number of events x. Under H₀, we assume X ~ Poisson(λ₀). We then calculate either the probability of observing x or more extreme values, or we compare x with critical values derived from the Poisson(λ₀) distribution.
检验统计量就是观测到的事件数 x。在 H₀ 下，我们假设 X ~ Poisson(λ₀)。然后我们计算观测到 x 或更极端值的概率，或者将 x 与从 Poisson(λ₀) 分布得到的临界值进行比较。
For the Poisson model to be valid, events must occur independently at a constant average rate, and two events cannot occur at exactly the same instant. In exam questions, these conditions are usually stated or assumed.
要使泊松模型有效，事件必须独立发生，平均发生率恒定，并且两个事件不能在同一瞬间发生。在考试题目中，这些条件通常会被说明或默认成立。
One important consequence of the Poisson variance being equal to the mean is that a single observed count x contains all the information about λ. There is no separate sample size n, because the interval length is fixed by the context of the question.
泊松分布的方差等于均值，这一重要性质意味着单个观测计数 x 包含了关于 λ 的全部信息。这里没有单独的样本容量 n，因为间隔长度由题目的上下文固定。
4. Critical Region Method | 临界区域法
In the critical region method, we find values of X that would lead us to reject H₀ at a given significance level α. For an upper-tail test, we look for the smallest integer c such that P(X ≥ c | λ = λ₀) ≤ α. The critical region is then X ≥ c.
在临界区域法中，我们找出在给定显著性水平 α 下会导致拒绝 H₀ 的 X 值。对于上尾检验，我们寻找最小的整数 c，使得 P(X ≥ c | λ = λ₀) ≤ α。临界区域则为 X ≥ c。
For a lower-tail test, we look for the largest integer c such that P(X ≤ c | λ = λ₀) ≤ α. The critical region is then X ≤ c. If the observed value x falls in the critical region, we reject H₀; otherwise, we do not reject H₀.
对于下尾检验，我们寻找最大的整数 c，使得 P(X ≤ c | λ = λ₀) ≤ α。临界区域则为 X ≤ c。如果观测值 x 落在临界区域内，我们拒绝 H₀；否则，不拒绝 H₀。
Because the Poisson distribution is discrete, the actual significance level of the test is usually smaller than α. For example, if α = 0.05 and λ₀ = 5, then P(X ≥ 10) = 0.0318 but P(X ≥ 9) = 0.0681, so we use c = 10, giving an actual significance level of 3.18%.
由于泊松分布是离散的，检验的实际显著性水平通常小于 α。例如，如果 α = 0.05 且 λ₀ = 5，则 P(X ≥ 10) = 0.0318 但 P(X ≥ 9) = 0.0681，因此我们使用 c = 10，实际显著性水平为 3.18%。
5. P-value Method | p 值法
The p-value is the probability of obtaining a result at least as extreme as the observed value x, assuming H₀ is true. For an upper-tail test, the p-value is P(X ≥ x | λ = λ₀). For a lower-tail test, it is P(X ≤ x | λ = λ₀).
p 值是在 H₀ 成立的前提下，得到与观测值 x 同等极端或更极端结果的概率。对于上尾检验，p 值为 P(X ≥ x | λ = λ₀)。对于下尾检验，p 值为 P(X ≤ x | λ = λ₀)。
We reject H₀ if the p-value is less than or equal to the significance level α. Many Edexcel questions ask for the p-value explicitly, so it is useful to practise both the critical region and p-value approaches.
如果 p 值小于或等于显著性水平 α，我们就拒绝 H₀。许多 Edexcel 题目明确要求计算 p 值，因此熟练运用临界区域法和 p 值法都很有用。
When using calculators such as the Casio ClassWiz, the Poisson cumulative distribution function (CDF) can be used directly to obtain p-values. Always set the mean parameter to λ₀ and read the appropriate cumulative probability carefully.
使用卡西欧 ClassWiz 等计算器时，可以直接使用泊松累积分布函数 (CDF) 求 p 值。务必把均值参数设为 λ₀，并仔细读取相应的累积概率。
6. One-tailed and Two-tailed Tests | 单尾与双尾检验
A one-tailed test uses all of the significance level α in one tail. An upper-tail test rejects H₀ for large observed counts, while a lower-tail test rejects H₀ for small observed counts. The alternative hypothesis indicates the direction.
单尾检验将显著性水平 α 全部用于一个尾部。上尾检验对大观测计数拒绝 H₀，而下尾检验对小观测计数拒绝 H₀。备择假设指明了方向。
For a two-tailed test with significance level α, the usual approach in Edexcel is to split α equally between the two tails. However, because the distribution is discrete, the two-tail probabilities are not always equal. One common method is to find a lower critical value c₁ and an upper critical value c₂ such that P(X ≤ c₁) ≤ α/2 and P(X ≥ c₂) ≤ α/2.
对于显著性水平为 α 的双尾检验，Edexcel 通常将 α 平均分配到两个尾部。但由于分布是离散的，两个尾部的概率并不总是相等。一种常见的方法是寻找下临界值 c₁ 和上临界值 c₂，使得 P(X ≤ c₁) ≤ α/2 且 P(X ≥ c₂) ≤ α/2。
Alternatively, the p-value for a two-tailed test can be calculated as p-value = 2 × min{P(X ≤ x), P(X ≥ x)}. This method is simple and avoids the need to construct a symmetric critical region. Students should be familiar with both methods.
另一种方法是，双尾检验的 p 值可以计算为 p-value = 2 × min{P(X ≤ x), P(X ≥ x)}。这种方法简单，避免了构造对称临界区域的麻烦。学生应熟悉这两种方法。
7. Using Poisson Cumulative Distribution Tables | 使用泊松累积分布表
In the exam, you may be given extracts from Poisson cumulative distribution tables, or you may need to use a calculator. The table gives values of P(X ≤ k) for different values of λ. To find an upper-tail probability, use P(X ≥ k) = 1 − P(X ≤ k − 1).
考试中可能会提供泊松累积分布表的摘录，或者允许使用计算器。表格给出了不同 λ 值下的 P(X ≤ k)。要求上尾概率，使用 P(X ≥ k) = 1 − P(X ≤ k − 1)。
When looking up critical values, first identify the λ₀ column. Then find the smallest k such that the cumulative probability exceeds 1 − α for an upper-tail test, or the largest k such that the cumulative probability is at most α for a lower-tail test.
查临界值时，首先找到 λ₀ 所在的列。对于上尾检验，找到使累积概率超过 1 − α 的最小 k；对于下尾检验，找到使累积概率不超过 α 的最大 k。
| λ = 5 | P(X ≤ k) | P(X ≥ k) |
|---|---|---|
| k = 7 | 0.8666 | P(X ≥ 8) = 0.1334 |
| k = 8 | 0.9319 | P(X ≥ 9) = 0.0681 |
| k = 9 | 0.9682 | P(X ≥ 10) = 0.0318 |
In the table above for λ = 5, if α = 0.05 and we are testing H₁: λ > 5, the critical value is c = 10 because P(X ≥ 10) = 0.0318 ≤ 0.05, but P(X ≥ 9) = 0.0681 > 0.05.
在上表中，对于 λ = 5，如果 α = 0.05 且我们检验 H₁: λ > 5，临界值为 c = 10，因为 P(X ≥ 10) = 0.0318 ≤ 0.05，而 P(X ≥ 9) = 0.0681 > 0.05。
8. Worked Example: One-tailed Upper Test | 例题：单尾上侧检验
A call centre claims that the average number of emergency calls received in a 10-minute period is 4. A manager suspects that the rate has increased. In a randomly chosen 10-minute period, 7 calls are received. Test at the 5% significance level whether the mean number of calls has increased.
某呼叫中心声称，10 分钟内接到的紧急呼叫平均次数为 4。一位经理怀疑该比率已上升。在随机选取的一个 10 分钟时间段内，接到了 7 次呼叫。在 5% 显著性水平下检验平均呼叫次数是否增加。
Let X be the number of calls in a 10-minute period. Hypotheses: H₀: λ = 4, H₁: λ > 4. Under H₀, X ~ Poisson(4). The observed value is x = 7. Using the p-value method, we calculate P(X ≥ 7 | λ = 4).
设 X 为 10 分钟内的呼叫次数。假设：H₀: λ = 4，H₁: λ > 4。在 H₀ 下，X ~ Poisson(4)。观测值为 x = 7。使用 p 值法，计算 P(X ≥ 7 | λ = 4)。
P(X ≥ 7) = 1 − P(X ≤ 6) = 1 − 0.8893 = 0.1107
Since the p-value 0.1107 is greater than 0.05, we do not reject H₀. There is insufficient evidence at the 5% level to conclude that the mean number of calls has increased.
由于 p 值 0.1107 大于 0.05，我们不拒绝 H₀。在 5% 显著性水平下，没有足够证据表明平均呼叫次数已增加。
If instead we used the critical region method, we would find the smallest c such that P(X ≥ c | λ = 4) ≤ 0.05. For λ = 4, P(X ≥ 8) = 0.0511 and P(X ≥ 9) = 0.0214, so c = 9. Since 7 < 9, the observed value is not in the critical region, and we do not reject H₀.
如果改用临界区域法，我们会找出使 P(X ≥ c | λ = 4) ≤ 0.05 的最小 c。对于 λ = 4，P(X ≥ 8) = 0.0511 且 P(X ≥ 9) = 0.0214，因此 c = 9。因为 7 < 9，观测值不在临界区域内，我们不拒绝 H₀。
9. Worked Example: Two-tailed Test | 例题：双尾检验
A machine is set to produce flaws in fabric at an average rate of 3 flaws per 10 square metres. To check whether the machine is still working correctly, a quality inspector examines a random 10 square metre piece and finds 8 flaws. Test at the 10% significance level whether the mean rate has changed.
一台机器设定为每 10 平方米织物平均产生 3 个瑕疵。为了检查机器是否仍正常工作，质检员随机检查了一块 10 平方米的织物，发现 8 个瑕疵。在 10% 显著性水平下检验平均发生率是否发生变化。
Let X be the number of flaws in 10 square metres. Hypotheses: H₀: λ = 3, H₁: λ ≠ 3. Under H₀, X ~ Poisson(3). The observed value is x = 8. For a two-tailed test, we can compute the p-value as p-value = 2 × min{P(X ≤ 8), P(X ≥ 8)}.
设 X 为 10 平方米内的瑕疵数。假设：H₀: λ = 3，H₁: λ ≠ 3。在 H₀ 下，X ~ Poisson(3)。观测值为 x = 8。对于双尾检验，p 值计算为 p-value = 2 × min{P(X ≤ 8), P(X ≥ 8)}。
P(X ≤ 8) = 0.9962, P(X ≥ 8) = 1 − P(X ≤ 7) = 1 − 0.9881 = 0.0119
The minimum of these two probabilities is 0.0119, so p-value = 2 × 0.0119 = 0.0238. Since 0.0238 < 0.10, we reject H₀. There is evidence at the 10% significance level that the mean rate of flaws has changed.
这两个概率中较小的是 0.0119，因此 p 值 = 2 × 0.0119 = 0.0238。由于 0.0238 < 0.10，我们拒绝 H₀。在 10% 显著性水平下，有证据表明平均瑕疵率发生了变化。
Using the critical region method with α/2 = 0.05 in each tail, we look for c₁ and c₂ such that P(X ≤ c₁) ≤ 0.05 and P(X ≥ c₂) ≤ 0.05. For λ = 3, P(X ≤ 0) = 0.0498, so c₁ = 0. For the upper tail, P(X ≥ 7) = 0.0335 and P(X ≥ 6) = 0.0839, so c₂ = 7. The critical region is X ≤ 0 or X ≥ 7. Since x = 8 is in the critical region, we reject H₀, agreeing with the p-value method.
使用临界区域法，每个尾部 α/2 = 0.05，寻找 c₁ 和 c₂ 使得 P(X ≤ c₁) ≤ 0.05 且 P(X ≥ c₂) ≤ 0.05。对于 λ = 3，P(X ≤ 0) = 0.0498，故 c₁ = 0。对于上尾，P(X ≥ 7) = 0.0335 且 P(X ≥ 6) = 0.0839，故 c₂ = 7。临界区域为 X ≤ 0 或 X ≥ 7。由于 x = 8 在临界区域内，我们拒绝 H₀，与 p 值法一致。
10. Conclusion and Common Pitfalls | 结论与常见误区
Testing for the mean of a Poisson distribution is a core skill in Edexcel A-Level Statistics. The main steps are: define the random variable, state the hypotheses, identify the distribution under H₀, calculate the p-value or compare with critical values, and write a conclusion in the context of the question.
泊松分布均值的假设检验是 Edexcel A-Level 统计学的一项核心技能。主要步骤是：定义随机变量、陈述假设、确定 H₀ 下的分布、计算 p 值或与临界值比较，并结合题目背景写出结论。
Common pitfalls include: using the wrong tail probability, forgetting to subtract 1 when finding P(X ≥ k), mixing up λ₀ with the observed count, and using a one-tailed test when the question asks for a two-tailed test. Always read the wording carefully.
常见误区包括：使用了错误的尾部概率、求 P(X ≥ k) 时忘记减 1、混淆 λ₀ 与观测计数，以及在题目要求双尾检验时误用单尾检验。务必仔细阅读题目措辞。
Another common issue is rejecting H₀ too easily when the observed count is only slightly above the expected mean. Because the Poisson distribution is skewed, large counts can occur by chance more often than intuition suggests. Always rely on the calculated p-value or critical region rather than personal judgement.
另一个常见问题是，当观测计数仅略高于期望均值时就轻易拒绝 H₀。由于泊松分布是偏斜的，较大的计数偶然出现的频率往往高于直觉。始终依据计算出的 p 值或临界区域，而不是个人判断。
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