Finding Critical Regions for a Geometric Distribution | 几何分布关键区域的确定

📚 Finding Critical Regions for a Geometric Distribution | 几何分布关键区域的确定

In Edexcel A-Level Mathematics, hypothesis tests for a geometric distribution require you to identify the set of sample values that lead to rejection of the null hypothesis. This article explains how to find critical regions for one-tailed and two-tailed tests using the cumulative probabilities of Geo(p).

在 Edexcel A-Level 数学中，几何分布的假设检验要求你找出导致拒绝原假设的样本值集合。本文讲解如何利用 Geo(p) 的累积概率来求单尾和双尾检验的关键区域。

1. Geometric Distribution Recap | 几何分布回顾

If X ~ Geo(p), then X counts the number of trials up to and including the first success. Its probability mass function is P(X = x) = p(1 − p)^(x − 1) for x = 1, 2, 3, … .

如果 X ~ Geo(p)，那么 X 表示直到首次成功为止的试验次数。其概率质量函数为 P(X = x) = p(1 − p)^(x − 1)，其中 x = 1, 2, 3, … 。

The two cumulative formulas needed for critical regions are P(X ≤ x) = 1 − (1 − p)^x and P(X ≥ x) = (1 − p)^(x − 1).

求关键区域所需的两条累积概率公式是 P(X ≤ x) = 1 − (1 − p)^x 和 P(X ≥ x) = (1 − p)^(x − 1)。

2. Hypothesis Test Structure | 假设检验结构

For a geometric distribution test, the null hypothesis is written as H₀: p = p₀, where p₀ is a specified value. The alternative hypothesis can be H₁: p < p₀, H₁: p > p₀, or H₁: p ≠ p₀.

对于几何分布检验，原假设写作 H₀: p = p₀，其中 p₀ 是一个给定的值。备择假设可以是 H₁: p < p₀、H₁: p > p₀ 或 H₁: p ≠ p₀。

The test statistic is the observed number of trials x until the first success. You compare this observed value with the critical region.

检验统计量是观察到首次成功所需的试验次数 x。你将这个观测值与关键区域进行比较。

3. From Alternative Hypothesis to Tail Direction | 从备择假设判断尾部方向

If H₁: p < p₀, a smaller probability of success means success is rarer, so the first success tends to occur later. Larger values of X are evidence against H₀, so the critical region is an upper tail of the form X ≥ c.

如果 H₁: p < p₀，成功的概率更小意味着成功更稀有，因此首次成功往往会更晚出现。较大的 X 值是对 H₀ 不利的证据，所以关键区域是形如 X ≥ c 的上尾。

If H₁: p > p₀, success is more common, so the first success tends to occur earlier. Smaller values of X are evidence against H₀, so the critical region is a lower tail of the form X ≤ c.

如果 H₁: p > p₀，成功更常见，因此首次成功往往出现得更早。较小的 X 值是对 H₀ 不利的证据，所以关键区域是形如 X ≤ c 的下尾。

If H₁: p ≠ p₀, both very small and very large values of X are suspicious, so you find two tails.

如果 H₁: p ≠ p₀，非常小和非常大的 X 值都值得怀疑，因此需要找出两个尾部。

4. Critical Region Defined | 关键区域的定义

The critical region is the set of values of X for which H₀ is rejected. At significance level α, the total probability of the critical region under H₀ must be no more than α.

关键区域是使得 H₀ 被拒绝的 X 值集合。在显著性水平 α 下，关键区域在原假设成立时的总概率不得超过 α。

Because a geometric distribution is discrete, the exact significance level α is rarely achieved. You choose the largest possible critical region whose probability is still ≤ α.

由于几何分布是离散的，精确达到显著性水平 α 的情况很少。你要选择概率仍不超过 α 的最大可能关键区域。

5. Upper-Tail Tests (H₁: p < p₀) | 上尾检验（H₁: p < p₀）

For an upper-tail test, reject H₀ if X ≥ c. You need the smallest integer c such that P(X ≥ c) ≤ α. Use P(X ≥ c) = (1 − p₀)^(c − 1).

对于上尾检验，如果 X ≥ c 就拒绝 H₀。你需要找到满足 P(X ≥ c) ≤ α 的最小整数 c。使用 P(X ≥ c) = (1 − p₀)^(c − 1)。

This means solving (1 − p₀)^(c − 1) ≤ α for c, usually by taking logs. If c is not an integer, round up to the next integer.

这意味着对 c 解不等式 (1 − p

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