Finding Critical Regions for a Poisson Distribution | 寻找泊松分布的临界区域

📚 Finding Critical Regions for a Poisson Distribution | 寻找泊松分布的临界区域

In Edexcel A-Level Mathematics, hypothesis testing using the Poisson distribution requires identifying critical regions - sets of values for the test statistic that lead to rejection of the null hypothesis. This article explains how to find these critical regions for one-tailed and two-tailed tests, carefully handling the discrete nature of the Poisson distribution.

在 Edexcel A-Level 数学中，使用泊松分布的假设检验需要确定临界区域 - 即会导致拒绝原假设的检验统计量取值集合。本文解释如何为单尾检验和双尾检验找到这些临界区域，并谨慎处理泊松分布的离散性。

The critical region is also called the rejection region. If the observed value falls inside it, we reject H₀ at the chosen significance level; otherwise, we do not reject H₀. The method is closely linked to the probability mass function of the Poisson distribution and to cumulative Poisson probabilities.

临界区域也称为拒绝域。如果观测值落入其中，我们在选定显著性水平下拒绝 H₀；否则不拒绝 H₀。该方法与泊松分布的概率质量函数以及累积泊松概率密切相关。

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1. Introduction to Hypothesis Testing and Critical Regions | 假设检验与临界区域简介

Hypothesis testing is a statistical method for making decisions about population parameters based on sample data. A critical region, or rejection region, is the set of values of the test statistic for which the null hypothesis H₀ is rejected. For a Poisson distribution, the test statistic is typically the observed count X from a Poisson process with rate parameter λ.

假设检验是一种基于样本数据对总体参数做出决策的统计方法。临界区域，或称拒绝域，是指检验统计量的一组取值，当检验统计量落入该区域时，拒绝原假设 H₀。对于泊松分布，检验统计量通常是从参数为 λ 的泊松过程中观测到的计数 X。

The significance level α, for example 0.05, determines the probability of rejecting H₀ when it is true. This is called a Type I error. The critical region is chosen so that the probability of the test statistic falling in that region is as close as possible to α, but without exceeding it for discrete distributions.

显著性水平 α，例如 0.05，决定了当 H₀ 为真时拒绝 H₀ 的概率。这称为第一类错误。临界区域的选择应使得检验统计量落入该区域的概率尽可能接近 α，但由于离散分布，不能超过 α。

P(X ∈ critical region | H₀) ≤ α

This inequality is the key rule for discrete distributions. Because X can only take integer values, we often cannot achieve exactly α. Instead, we choose the largest or smallest integer boundary that keeps the total rejection probability at or below α.

这个不等式是离散分布的关键规则。因为 X 只能取整数值，我们通常无法恰好达到 α。相反，我们选择最大或最小的整数边界，使总拒绝概率保持在不大于 α 的水平。

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2. The Poisson Distribution in Hypothesis Tests | 假设检验中的泊松分布

A common scenario involves a random variable X ~ lt; λ₀ for a lower-tail test, H₁: λ > λ₀ for an upper-tail test, or H₁: λ ≠ λ₀ for a two-tailed test.

常见情形涉及随机变量 X ~ Po(λ)，其中 λ 是已知或假设的平均发生率。我们检验关于 λ 的假设，例如 H₀: λ = λ₀ 对 H₁: λ < λ₀ 的下尾检验，H₁: λ > λ₀ 的上尾检验，或 H₁: λ ≠ λ₀ 的双尾检验。

n variables each with mean λ, then their sum S = X₁ + X₂ + ... + Xₙ has the distribution S ~ Po(nλ). In that case, the test statistic is usually S, and the null mean is nλ₀.

我们可能有一个单次观测 X = x，也可能有一组观测之和。如果 X₁, X₂, ..., Xₙ 是相互独立的泊松变量，每个均值为 λ，则它们的和 S = X₁ + X₂ + ... + Xₙ 服从分布 S ~ Po(nλ)。此时检验统计量通常为 S，原假设下的均值为 nλ₀。

The Poisson probability mass function is used to calculate probabilities of individual counts:

泊松概率质量函数用于计算单个计数的概率：

P(X = x) = e⁻λ λˣ / x!, x = 0, 1, 2, ...

For a critical region, we usually need cumulative probabilities rather than individual probabilities. The cumulative distribution function is:

对于临界区域，我们通常需要累积概率，而不是单个概率。累积分布函数为：

P(X ≤ r) = Σ (x = 0 to r) e⁻λ λˣ / x!

These cumulative probabilities are exactly what we compare with α or α/2 when finding critical values.

这些累积概率正是我们在寻找临界值时与 α 或 α/2 比较的对象。

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3. Setting Up Hypotheses and the Test Statistic | 建立假设与检验统计量

Before finding a critical region, you must define the random variable clearly. For example, let X be the number of emails arriving in one hour, and assume X ~ Po(λ). State H₀ and H₁ in terms of λ, and identify whether the test is one-tailed or two-tailed.

在寻找临界区域之前，必须清楚地定义随机变量。例如，设 X 为一小时内到达的电子邮件数量，并假设 X ~ Po(λ)。用 λ 写出 H₀ 和 H₁，并判断检验是单尾还是双尾。

The null hypothesis usually gives a specific value λ₀. The alternative hypothesis determines the direction of the critical region. A lower-tail test uses small values of X as evidence against H₀. An upper-tail test uses large values of X. A two-tailed test uses both very small and very large values.

原假设通常给出一个具体值 λ₀。备择假设决定临界区域的方向。下尾检验使用较小的 X 值作为反对 H₀ 的证据。上尾检验使用较大的 X 值。双尾检验同时使用非常小和非常大的值。

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For a one-tailed test at significance level α, the whole α is placed in one tail. For a two-tailed test at significance level α, the probability is usually split equally, with α/2 in each tail. This split is a convention, but it is the standard approach in Edexcel A-Level Mathematics.

对于显著性水平为 α 的单尾检验，整个 α 放在一个尾部。对于显著性水平为 α 的双尾检验，概率通常等分，每个尾部为 α/2。这种分配是一种约定，但它是 Edexcel A-Level 数学中的标准做法。

You should also decide whether the test statistic is a single count X or a total count S. If the data are collected over n independent periods or regions, the total S ~ Po(nλ) under H₀ with mean nλ₀.

你还应决定检验统计量是单个计数 X 还是总计数 S。如果数据来自 n 个独立时期或区域，则在 H₀ 下总计数 S ~ Po(nλ)，均值为 nλ₀。

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4. Lower-Tail Critical Regions | 下尾临界区域

s the form X ≤ c, where c is an integer. We choose the largest integer c such that P(X ≤ c | H₀) ≤ α.

下尾检验的 H₁: λ < λ₀。当 X 异常小时拒绝 H₀。因此临界区域具有 X ≤ c 的形式，其中 c 是整数。我们选择最大的整数 c，使得 P(X ≤ c | H₀) ≤ α。

α. If P(X ≤ c) ≤ α but P(X ≤ c + 1) > α, then the critical region is {0, 1, 2, ..., c}.

实际操作中，计算累积概率 P(X ≤ 0), P(X ≤ 1), P(X ≤ 2), ...，找到最后一个仍然不超过 α 的概率。如果 P(X ≤ c) ≤ α 但 P(X ≤ c + 1) > α，则临界区域为 {0, 1, 2, ..., c}。

The actual significance level of the test is P(X ≤ c | and P(X ≤ 0) = 0.0183 while P(X ≤ 1) = 0.0916, then c = 0 and the actual level is 0.0183.

检验的实际显著性水平为 P(X ≤ c | H₀)，它可能小于 α。例如，若 α = 0.05，P(X ≤ 0) = 0.0183，而 P(X ≤ 1) = 0.0916，则 c = 0，实际水平为 0.0183。

tail test, write X ≤ c or {X : X ≤ c}. The conclusion should always be in context, such as 'there is sufficient evidence at the 5% level that the mean rate is less than λ₀'.

书写临界区域时，使用集合符号。对于下尾检验，写 X ≤ c 或 {X : X ≤ c}。结论应始终结合情境，例如 '在 5% 水平上有充分证据表明平均发生率小于 λ₀'。

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5. Upper-Tail Critical Regions | 上尾临界区域

The critical region has the form X ≥ c. We choose the smallest integer c such that P(X ≥ c | H₀) ≤ α.

上尾检验的 H₁: λ > λ₀。当 X 异常大时拒绝 H₀。临界区域具有 X ≥ c 的形式。我们选择最小的整数 c，使得 P(X ≥ c | H₀) ≤ α。

For a candidate value c, calculate P(X ≥ c) = 1 - P(X ≤ c - 1). Then compare this with α.

因为大多数累积泊松表给出 P(X ≤ r)，通常更容易使用补概率。对于候选值 c，计算 P(X ≥ c) = 1 - P(X ≤ c - 1)。然后将其与 α 比较。

. This form is useful when working from a cumulative table. The critical region is then {c, c + 1, c + 2, ...}.

等价地，我们需要 P(X ≤ c - 1) ≥ 1 - α。当使用累积表时，这种形式很有用。临界区域则为 {c, c + 1, c + 2, ...}。

Again, the actual significance level is P(X ≥ c | stribution means that we cannot usually make the rejection probability exactly equal to α.

同样，实际显著性水平为 P(X ≥ c | H₀)，它可能低于 α。泊松分布的离散性意味着我们通常无法使拒绝概率恰好等于 α。

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6. Two-Tailed Critical Regions | 双尾临界区域

of X. The critical region is the union of a lower-tail part and an upper-tail part: X ≤ c_L or X ≥ c_U.

双尾检验的 H₁: λ ≠ λ₀。当 X 非常小或非常大时拒绝 H₀。临界区域是下尾部分和上尾部分的并集：X ≤ c_L 或 X ≥ c_U。

cance level α, split the probability equally: α/2 in each tail. Find the largest integer c_L such that P(X ≤ c_L | H₀) ≤ α/2. Find the smallest integer c_U such that P(X ≥ c_U | H₀) ≤ α/2. The critical region is {0, 1, ..., c_L} ∪ {c_U, c_U + 1, ...}.

对于显著性水平 α，将概率等分：每个尾部 α/2。找到最大的整数 c_L，使得 P(X ≤ c_L | H₀) ≤ α/2。找到最小的整数 c_U，使得 P(X ≥ c_U | H₀) ≤ α/2。临界区域为 {0, 1, ..., c_L} ∪ {c_U, c_U + 1, ...}。

The total actual significance level is P(X ≤ c_L | H₀) + P(X ≥ c_U | ₀). This total may be less than α because both tails are discrete. It should not exceed α.

总实际显著性水平为 P(X ≤ c_L | H₀) + P(X ≥ c_U | H₀)。由于两个尾部都是离散的，这个总和可能小于 α。它不应超过 α。

Some students try to find a single value that splits the distribution into two equal tails. That is not correct for a Poisson test. You must work with the cumulative probabilities on each side separately.

有些学生试图找到一个值，将分布分成两个相等的尾部。对于泊

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