📚 Finding Critical Regions for a Poisson Distribution | 泊松分布的临界域确定
When you carry out a hypothesis test for a Poisson mean, the critical region tells you exactly which observed counts should lead you to reject H₀. This article shows how to find critical regions for one-tailed and two-tailed Poisson tests at a given significance level, how to report the actual significance level, and how to avoid the mistakes that cost marks most often.
当你对泊松均值进行假设检验时，临界域会告诉你哪些观测计数会导致拒绝原假设 H₀。本文介绍如何在给定显著性水平下，为单尾和双尾泊松检验确定临界域，如何报告实际显著性水平，以及如何避开最常见的失分点。
1. Recap: The Poisson Distribution | 回顾：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, when events happen independently and at a constant average rate. If X ~ Po(λ), then the mean and variance are both λ, so a single parameter controls the whole distribution.
泊松分布用于对固定时间或空间区间内事件发生次数进行建模，条件是事件独立发生且平均发生率恒定。若 X ~ Po(λ)，则均值与方差均为 λ，因此整个分布只由一个参数控制。
P(X = x) = e⁻λ λˣ / x! , x = 0, 1, 2, ...
The formula uses e⁻λ as the exponential term, λˣ as λ raised to the power x, and x! e key diagnostic: if a real data set has a variance much larger than its mean, a Poisson model is probably unsuitable.
公式中 e⁻λ 是自然指数项，λˣ 表示 λ 的 x 次方，x! 是 x 的阶乘。均值与方差满足 E(X) = Var(X) = λ 是关键判别条件：如果真实数据的方差远大于均值，泊松模型通常不合适。
use. For example, the lower-tail probability is built up term by term starting from x = 0.
累积概率通过求和得到，而假设检验真正使用的正是累积概率。例如下尾概率从 x = 0 开始逐项累加得到。
P(X ≤ k) = Σₓ₌₀ᵏ e⁻λ λˣ / x!
umulative tables or from a calculator function such as Poisson CDF, rather than summing by hand.
在 A-Level 考试中，你通常直接从泊松累积概率表或计算器的泊松累积分布函数（Poisson CDF）读取 P(X ≤ k)，而不必手算求和。
2. What Is a Critical Region? | 什么是临界域？
A critical region, also called a rejection region, is the set of values of the test statistic for which the null hypothesis is rejected. The boundary values are called critical values. If the observed test statistic falls in the critical region, the result is significant at the chosen level.
临界域，也称为拒绝域，是检验统计量的取值集合，当统计量落入该集合时拒绝原假设。边界值称为临界值。如果观测检验统计量落入临界域，则结果在该选定水平下显著。
itical region is therefore a range of counts such as X ≤ c, X ≥ c, or both tails together.
在泊松假设检验中，检验统计量就是观测计数 X，来自一个区间或一组区间。因此临界域是一个计数范围，例如 X ≤ c、X ≥ c，或两个尾部同时出现。
The critical region is defined before the data are collected, using the distribution of X under H₀. That is essential: choosing the cut-off after seeing the data would invalidate the test.
临界域必须在收集数据之前、利用 H₀ 下 X 的分布来确定。这一点至关重要：先看数据再选临界值会破坏检验的有效性。
The size of the critical region is the probability of rejecting H₀ when H₀ is true, which is exactly the significance level α when the test is exact.
临界域的大小就是当 H₀ 为真时拒绝 H₀ 的概率，当检验恰好达到名义水平时，它正好等于显著性水平 α。
3. Hypothesis Test Setup for Poisson | 泊松分布假设检验设置
For a Poisson test, the null hypothesis usually states that the population mean λ equals a known or claimed value λ₀. The alternative hypothesis determines whether the test is lower-tail, upper-tail, or two-tailed.
对于泊松检验，原假设通常表述为总体均值 λ 等于已知值或声称值 λ₀。备择假设决定检验是下限单尾、上限单尾还是双尾。
| Alternative hypothesis | Tail | 中文判断 |
|---|---|---|
| H₁: λ < λ₀ | Lower-tail | X 过小时拒绝 |
| H₁: λ > λ₀ | Upper-tail | X 过大时拒绝 |
| H₁: λ ≠ λ₀ | Two-tailed | X 过小或过大都拒绝 |
The wording of the question normally reveals the direction. Phrases such as "has increased", "is higher than before" or "has risen" point to an upper-tail test, while "has decreased" or "has fallen" points to a lower-tail test.
题目的措辞通常就能揭示方向。诸如 "has increased"、"is higher than before"、"has risen"（有所上升）之类的表述指向上限单尾检验；而 "has decreased"、"has fallen"（有所下降）则指向下限单尾检验。
One technical point matters here: the Poisson parameter λ refers to the mean count per interval, so if the interval length changes, λ must be rescaled before you set up the test.
这里有一个技术要点：泊松参数 λ 指的是每个区间内的平均计数，因此如果区间长度改变，必须先对 λ 做相应缩放，再建立检验。
4. Finding a Lower-Tail Critical Region | 求下限单尾临界域
or example 0.05). Small values of X support H₁, so the critical region has the form X ≤ c.
设 H₀: λ = λ₀ 对 H₁: λ < λ₀，显著性水平为 α（用小数表示，例如 0.05）。X 取值小支持 H₁，因此临界域形如 X ≤ c。
The rule is to choose the largest integer c such that the probability of the region does not exceed α:
规则是：选取使临界域概率不超过 α 的最大整数 c：
P(X ≤ c | λ = λ₀) ≤ α
becomes the critical value, and the critical region is X ≤ c or, when c = 0, simply X = 0.
从 c = 0 开始，逐步增大 c，直到再加入下一个取值会使累积概率超过 α。越界前的那一个值即为临界值，临界域为 X ≤ c；若 c = 0，则临界域就是单点 X = 0。
ill inside the region, so a useful self-check is that P(X ≤ c + 1) must be greater than α.
注意临界值 c 是仍在域内的最大计数，因此一个实用的自检方法是：P(X ≤ c + 1) 必须大于 α。
most always be strictly less than α. The actual significance level is exactly that value, and examiners often ask for it.
由于 X 是离散的，P(X ≤ c) 几乎总是严格小于 α。实际显著性水平就是该值，考官经常会要求你写出它。
5. Finding an Upper-Tail Critical Region | 求上限单尾临界域
t; λ₀ at level α. Large values of X support H₁, so the critical region has the form X ≥ c.
现在设 H₀: λ = λ₀ 对 H₁: λ > λ₀，水平为 α。X 取值大支持 H₁，因此临界域形如 X ≥ c。
The working rule is to choose the smallest integer c such that:
操作方法为：选取使下式成立的最小整数 c：
P(X ≥ c | λ = λ₀) ≤ α
+ 1. This conversion is worth practising because most tables only tabulate the lower tail.
等价地，利用累积概率表，先找到满足 P(X ≤ k) ≥ 1 − α 的最大整数 k，则 c = k + 1。这一转换值得反复练习，因为多数概率表只给出下尾数值。
equal to α, you have cut the region too small and lost a value that should have been included.
还要检查相邻取值。如果 P(X ≥ c − 1) 仍然不超过 α，说明你把临界域切得太小，漏掉了本应包含的取值。
rm P(X = c), which can easily be the difference between a correct and an incorrect answer.
常见失误是算成 P(X > c) 而不是 P(X ≥ c)。对于整数计数，两者恰好相差一项 P(X = c)，而这往往就是答案对错的分水岭。
6. Two-Tailed Tests and Splitting the Significance Level | 双尾检验与显著性水平的分配
e counts are evidence against H₀, so the critical region has two parts: X ≤ c₁ and X ≥ c₂.
对于 H₁: λ ≠ λ₀，过小和过大的计数都不利于 H₀，因此临界域由两部分组成：X ≤ c₁ 与 X ≥ c₂。
. If α = 0.10 then each tail is allowed 0.05; if α = 0.05 then each tail is allowed 0.025.
在 A-Level 中，最稳妥且被普遍接受的约定是把显著性水平平均分配，即每尾各占 α/2。若 α = 0.10，则每尾各允许 0.05；若 α = 0.05，则每尾各允许 0.025。
You then apply the rules from Sections 4 and 5 independently, using α/2 in place of α in each tail.
然后分别独立地套用第 4 节和第 5 节的规则，只是每尾用 α/2 代替 α。
An alternative convention places all of α on whichever tail produces the smaller probability. Both approaches appear in textbooks, so read the question; if it says "each tail", "symmetric" or "split equally", use the equal split.
另一种约定是把全部 α 放在概率更小的那一尾。两种做法在教材中都能见到，所以务必阅读题干；如果题目出现 "each tail"、"symmetric" 或 "split equally"（等分），就用等分法。
The actual significance level of a two-tailed test is the sum of the two tail probabilities, and it will be noticeably below α because of discreteness.
双尾检验的实际显著性水平是两个尾部概率之和，由于离散性，它会明显小于名义 α。
7. Using Tables and Calculators | 使用累积概率表与计算器
X ≤ k). Read along the row for λ = λ₀ until you find the entries that straddle α or 1 − α.
泊松累积概率表通常以 λ 为行、k 为列，给出 P(X ≤ k)。沿着 λ = λ₀ 那一行读取，找到跨过 α 或 1 − α 的那些数值。
Most modern syllabuses allow a calculator. On a typical graphical calculator you use Poisson CDF directly, entering the value of λ and the upper limit k.
大多数现代考纲允许使用计算器。在常见的图形计算器上，你直接调用泊松累积分布函数 Poisson CDF，输入 λ 值和上限 k 即可。
If your calculator only provides the probability mass function, build the cumulative value by repeated addition and keep a running total in a table. Writing the running total down makes the final decision transparent and earns method marks.
如果你的计算器只提供概率质量函数，就通过反复相加来构造累积概率，并在表格中记录累加值。把累加过程写下来能让最终判断一目了然，也更容易拿到过程分。
| Step | What to do | 中文说明 |
|---|---|---|
| 1 | State H₀ and H₁, and the value of λ₀ | 写出 H₀、H₁ 与 λ₀ |
| 2 | Identify the tail(s) from H₁ | 由 H₁ 判断单尾或双尾 |
| 3 | Use α, or α/2 per tail, as the allowance | 确定每尾允许的 α 或 α/2 |
| 4 | Find the extreme integer c where the tail probability first stays ≤ allowance | 找到尾部概率首次不超过允许值的极端整数 c |
| 5 | State the critical region and the actual significance level | 写出临界域与实际显著性水平 |
8. Worked Example: Upper-Tail Test | 例题精讲：上限单尾检验
A call centre receives calls at an average rate of 3 per hour, and X ~ A new system is claimed to increase the rate. Test H₀: λ = 3 against H₁: λ > 3 at the 5% level, using a single hour of observation, and state the critical region.
某呼叫中心平均每小时接到 3 通电话，H₀ 下假定 X ~ Po(3)。有人声称新系统会提高来电率。用一小时观测数据，在 5% 水平下检验 H₀: λ = 3 对 H₁: λ > 3，并写出临界域。
llest c with P(X ≥ c) ≤ 0.05. Building the lower-tail cumulative probabilities with λ = 3:
我们需要满足 P(X ≥ c) ≤ 0.05 的最小 c。取 λ = 3 逐步计算下尾累积概率：
| k | P(X ≤ k) | P(X ≥ k + 1) |
|---|---|---|
| 5 | 0.9161 | 0.0839 |
| 6 | 0.9665 | 0.0335 |
| 7 | 0.9881 | 0.0119 |
o the critical region is X ≥ 7, and the actual significance level is 0.0335, or about 3.35%.
P(X ≥ 7) = 0.0335 ≤ 0.05，而 P(X ≥ 6) = 0.0839 > 0.05，因此临界域为 X ≥ 7，实际显著性水平为 0.0335，约 3.35%。
If the observed count is 8 calls in the hour, then 8 lies in the critical region, so we reject H₀ and conclude there is sufficient evidence at the 5% level that the call rate has increased.
若该小时实际观测到 8 通来电，则 8 落在临界域内，因此拒绝 H₀，结论是在 5% 水平下有充分证据表明来电率有所上升。
9. Worked Example: Lower-Tail and Two-Tailed Cases | 例题精讲：下限单尾与双尾情形
For a lower-tail test, suppose H₀: λ = 4 against H₁: λ < 4 at the 5% level. With λ = 4, e⁻⁴ = 0.018316, so P(X = 0) = 0.0183 and P(X ≤ 1) = 0.0916.
对于下限单尾检验，设 H₀: λ = 4 对 H₁: λ < 4，水平 5%。当 λ = 4 时，e⁻⁴ = 0.018316，故 P(X = 0) = 0.0183，P(X ≤ 1) = 0.0916。
ctual significance level 0.0183. The test is quite conservative here, which is typical when λ₀ is small.
由于 0.0183 ≤ 0.05 而 0.0916 > 0.05，临界域为 X = 0，实际显著性水平 0.0183。此时检验相当保守，这在 λ₀ 较小时很常见。
Now take a two-tailed test with H₀: λ = 5 against H₁: λ ≠ 5 at the 10% level, so each tail allows 0.05. Using λ = 5 cumulative probabilities:
再看双尾检验，H₀: λ = 5 对 H₁: λ ≠ 5，水平 10%，故每尾允许 0.05。取 λ = 5 的累积概率：
P(X ≤ 1) = 0.0404 ≤ 0.05 ; P(X ≥ 11) = 0.0137 ≤ 0.05
re X ≤ 1 or X ≥ 11. The actual significance level is 0.0404 + 0.0137 = 0.0541, roughly 5.4%, comfortably below the nominal 10%.
因此临界域为 X ≤ 1 或 X ≥ 11。实际显著性水平为 0.0404 + 0.0137 = 0.0541，约 5.4%，远低于名义的 10%。
Notice how much the discreteness costs you here. A continuous test at 10% would use almost the whole allowance, whereas this discrete test uses only about half of it, making it harder to reject H₀.
注意这里的离散性代价有多大。连续型检验在 10% 水平下几乎会用满全部允许概率，而这个离散检验只用了大约一半，因而更难拒绝 H₀。
An observation of 12 would fall in the critical region and lead to rejection; an observation of 9 would not, even though 9 is above the mean, because it is not extreme enough.
若观测值为 12，会落入临界域并导致拒绝；若观测值为 9，则不会拒绝，尽管 9 高于均值，但还不够极端。
10. Actual Significance Level and Exam Wording | 实际显著性水平与考题表述
it is P(X ≤ c) or P(X ≥ c) under H₀; for a two-tailed test it is the sum of the two tails.
实际（或达到的）显著性水平是你所选临界域对应的真实第一类错误概率。对单尾检验，它是 H₀ 下的 P(X ≤ c) 或 P(X ≥ c)；对双尾检验，它是两个尾部概率之和。
It is always less than or equal to the nominal level α because the counts are discrete and you cannot hit
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