📚 Hypothesis Testing for the Parameter p of a Geometric Distribution | 几何分布参数 p 的假设检验
Hypothesis testing for the parameter p of a geometric distribution is a statistical procedure used to decide whether the true probability of success in a sequence of Bernoulli trials differs from a claimed value. In A-Level Mathematics under Edexcel, you will learn to set up null and alternative hypotheses, use the first success trial number as the test statistic, and calculate p‑values or critical regions based on the cumulative probabilities of a geometric distribution. This article will guide you through every key concept, from recapping the geometric model to step‑by‑step hypothesis tests, with careful attention to both one‑tailed and two‑tailed scenarios.
几何分布参数 p 的假设检验是一种统计推断方法，用于判断在一系列伯努利试验中，真实成功概率是否与某个宣称值不同。在爱德思 A‑Level 数学课程中，你需要学会建立原假设与备择假设，以第一次成功所需的试验次数作为检验统计量，并利用几何分布的累积概率计算 p‑值或确定临界区域。本文将带你走完每一个关键概念，从几何分布回顾到逐步假设检验示例，并重点关注单尾与双尾检验的区别。
1. Recap of the Geometric Distribution | 几何分布回顾
A random variable X is said to follow a geometric distribution with parameter p, written X ~ Geo(p), if it counts the number of trials up to and including the first success in a sequence of independent Bernoulli trials, each with success probability p. The probability mass function (PMF) is given by:
若随机变量 X 表示在一系列独立伯努利试验中，直到首次成功为止（含成功那次）的试验次数，且每次试验成功概率为 p，则称 X 服从参数为 p 的几何分布，记作 X ~ Geo(p)。其概率质量函数为：
P(X = x) = p(1 − p)ˣ⁻¹, x = 1, 2, 3, …
r, the probability that the first success occurs on or after trial x is P(X ≥ x) = (1 − p)ˣ⁻¹, and the probability that it occurs on or before trial x is P(X ≤ x) = 1 − (1 − p)ˣ. These forms follow from the geometric series properties and will be used to compute p‑values and critical regions.
关键的累积概率在假设检验中至关重要。特别地，首次成功发生在第 x 次或之后的概率为 P(X ≥ x) = (1 − p)ˣ⁻¹，而发生在第 x 次或之前的概率为 P(X ≤ x) = 1 − (1 − p)ˣ。这些公式源自等比数列求和性质，将用于计算 p‑值 和临界区域。
2. Principles of Hypothesis Testing | 假设检验的基本原理
th a null hypothesis H₀, which is the assumption that a parameter takes a particular value — suspect might be true instead: p < p₀, p > p₀, or p ≠ p₀. A significance level α (often 5% or 1% sen before the test. The observed data provide a test statistic, and we either compute a p‑value — ability of obtaining a result at least as extreme as the observed one, assuming H₀ is true — or locate the observed value inside a critical region. If the p‑lue ≤ α, or the test statistic falls in the critical region, we reject H₀ in favour of H₁.
假设检验从原假设 H₀ 开始，它假定参数取某个特定值——此处 H₀: p = p₀。备择假设 H₁ 陈述我们怀疑可能成立的替代情况：p < p₀, p > p₀, 或 p ≠ p₀。检验前要事先选定显著性水平 α（通常为 5% 或 1%）。观测数据给出检验统计量，我们或者计算 p‑值——即在 H₀ 为真的条件下，得到至少与实际观测一样极端的结果的概率——或者直接判断观测值是否落入临界区域。一旦 p‑值 ≤ α，或检验统计量落入临界区域，我们就拒绝 H₀，支持 H₁。
s discrete, the critical region is often defined by inequalities such as X ≥ c or X ≤ c, and the exact probability of the region may not equal α exactly; we choose c so that the probability is as close as possible to α without exceeding it.
在几何分布的场景中，检验统计量就是观测到的直到首次成功所需的试验次数，记作 x。由于分布是离散的，临界区域通常由形如 X ≥ c 或 X ≤ c 的不等式定义，且区域的实际概率可能无法恰好等于 α；我们选择 c 使得该概率尽可能接近 α 但不超过它。
3. The Parameter p in Context | 背景中的参数 p
ores a free throw, the chance that a component passes a quality test, or the likelihood a seed germinates. The null hypothesis typically reflects a claimed or historical value, e.g., H₀: p = 0.3. If we suspect the true success rate has decreased, we would set H₁: p < 0.3, meaning a smaller p would make it harder to get an early success, thereby tending to produce larger observed trial counts x. Conversely, a larger p would make success occur sooner, so H₁: p > 0.3 would lead us to look for unusually small values of x.
参数 p 表示每次试验的真实成功概率。常见的考试背景包括篮球运动员罚球命中的概率、元件通过质量检测的概率，或种子发芽的概率。原假设通常反映一个宣称值或历史值，例如 H₀: p = 0.3。如果我们怀疑真实成功率下降了，就会设 H₁: p < 0.3，这意味着更小的 p 会使得早期成功更难发生，从而倾向于产生更大的观测次数 x。相反，更大的 p 会使得成功更早出现，因此 H₁: p > 0.3 会引导我们关注异常小的 x 值。
4. Setting Up Hypotheses | 建立假设
p: H₀: p = p₀. The alternative hypothesis determines the direction of the test. For a one‑tailed lower‑tail test (suspecting p has fallen), write H₁: p < p₀. For an upper‑tail test (suspecting p has risen), H₁: p > p₀. For a two‑ . Always define the test statistic X as the number of trials until the first success, so X ~ Geo(p).
准确定义 H₀ 和 H₁ 是第一个正式步骤。原假设必须包含一个准确的 p 值：H₀: p = p₀。备择假设决定检验的方向。对于单尾下限检验（怀疑 p 变小了），用 H₁: p < p₀。对于单尾上限检验（怀疑 p 变大了），用 H₁: p > p₀。对于双尾检验（与宣称值有任何偏离），用 H₁: p ≠ p₀。始终要明确定义检验统计量 X 为直至首次成功的试验次数，即 X ~ Geo(p)。
5. Test Statistic: The Observed Count | 检验统计量：观测计数
. No further transformation is needed. For example, if a basketball player claims her free‑ attempts and finally scores on the 5th attempt, then the observed test statistic is x = 5.
一旦实际进行了试验并观测到首次成功正好发生在第 x 次试验，这个数字 x 就成为我们的检验统计量，无需再做任何变换。例如，某篮球运动员声称罚球命中率为 0.8，而我们在观测中看到她前四次都失手，直到第五次才命中，那么观测到的检验统计量就是 x = 5。
With X ~ Geo(p₀), we can compute tail probabilities under the assumption that H₀ is true. The choice of which tail to look at depends on H₁, as illustrated later.
在 X ~ Geo(p₀) 且 H₀ 为真的假设下，我们可以计算尾部概率。至于选择哪一侧尾部，取决于 H₁，我们将在后文说明。
6. One‑Tailed Testing: Lower‑Tail (H₁: p < p₀) | 单尾检验：左侧尾部 (H₁: p < p₀)
g times for the first success. Hence, large values of x provide evidence against H₀. The p‑ e direction supporting H₁, i.e., P(X ≥ x) under H₀. Using the geometric cumulative form, p‑value = (1 − p₀)ˣ⁻¹. If this p‑value ≤ α, we reject H₀.
当 H₁: p < p₀ 时，更小的成功概率会导致首次成功前的等待时间变长。因此，较大的 x 值构成反对 H₀ 的证据。p‑值就是在 H₀ 为真的条件下，观测到与 x 同等或更加倾向于 H₁ 的结果的概率，即 P(X ≥ x)。利用几何分布的累积形式，p‑值 = (1 − p₀)ˣ⁻¹。若此 p‑值 ≤ α，则拒绝 H₀。
For example, suppose H₀: p = 0.6, H₁: p < 0.6, α = 0.05. We observe x = 7. The p‑ 1 < 0.05, we reject H₀ and conclude that the true success probability is likely below 0.6.
例如，设 H₀: p = 0.6, H₁: p < 0.6, α = 0.05。我们观测到 x = 7。p‑值 = (1 − 0.6)⁷⁻¹ = 0.4⁶ ≈ 0.0041。由于 0.0041 < 0.05，我们拒绝 H₀，认为真实成功概率很可能低于 0.6。
7. One‑Tailed Testing: Upper‑Tail (H₁: p > p₀) | 单尾检验：右侧尾部 (H₁: p > p₀)
er success rate makes early successes more likely, so small values of x are extreme. The p‑value is then P(X ≤ x) under H₀, which equals 1 − (1 − p₀)ˣ. We reject H₀ if this p‑value ≤ α.
若备择假设为 H₁: p > p₀，更高的成功率会使早期成功更易出现，因此较小的 x 值才是极端的。此时 p‑值为在 H₀ 下 P(X ≤ x) = 1 − (1 − p₀)ˣ。若该 p‑值 ≤ α，则拒绝 H₀。
Consider H₀: p = 0.2, H₁: p > 0.2, α = 0.05, with observed x = 1. The p‑ e 0.2 > 0.05, we do not reject H₀. There is insufficient evidence against the claimed 0.2.
考虑 H₀: p = 0.2, H₁: p > 0.2, α = 0.05，并观测到 x = 1。p‑值 = 1 − (0.8)¹ = 0.2。因为 0.2 > 0.05，我们不拒绝 H₀，即没有足够证据否定宣称的 0.2。
8. Critical Region Approach | 临界区域法
Instead of computing a p‑ ere c is the smallest integer such that P(X ≥ c) ≤ α under H₀. That means we want (1 − p₀)ᶜ ritical region is X ≤ c, where c is the largest integer satisfying P(X ≤ c) = 1 − (1 − p₀)ᶜ , c ≤ ln(1 − α) / ln(1 − p₀). If the observed x falls into the critical region, reject H₀.
除了计算 p‑值，也可以在试验前确定临界区域。对于 H₁: p < p₀，临界区域形如 X ≥ c，其中 c 是满足在 H₀ 下 P(X ≥ c) ≤ α 的最小整数，即 (1 − p₀)ᶜ⁻¹ ≤ α。通过对数求解 c：c ≥ 1 + ln α / ln(1 − p₀)。对于 H₁: p > p₀，临界区域为 X ≤ c，其中 c 是满足 P(X ≤ c) = 1 − (1 − p₀)ᶜ ≤ α 的最大整数，整理可得 c ≤ ln(1 − α) / ln(1 − p₀)。若观测到的 x 落入临界区域，则拒绝 H₀。
ritical region explicitly. Always state the region in terms of the test statistic X, e.g., 'Critical region is X ≥ 8' or 'X ≤ 2'.
这种方法在要求明确写出临界区域的考试题中尤其有用。务必使用检验统计量 X 来表达区域，例如“临界区域为 X ≥ 8”或“X ≤ 2”。
9. Two‑Tailed Tests | 双尾检验
n the two tails, aiming for each tail probability ≤ α/2. For the lower tail (small x, suggesting p > p₀), find the largest c₁ such that P(X ≤ c₁) ≤ α/2. For the upper tail (large x, suggesting p < p₀), find the smallest c₂ such that P(X ≥ c₂) ≤ α/2. The critical region is then X ≤ c₁ or X ≥ c₂. If the observed x falls in either region, reject H₀.
当 H₁: p ≠ p₀ 时，反对 H₀ 的证据既可能来自很小的 x，也可能来自很大的 x。常见的做法是将显著性水平 α 平均分配到双尾，使每侧概率 ≤ α/2。对于下侧尾部（小 x 暗示 p > p₀），找出最大的 c₁ 满足 P(X ≤ c₁) ≤ α/2；对于上侧尾部（大 x 暗示 p < p₀），找出最小的 c₂ 满足 P(X ≥ c₂) ≤ α/2。临界区域即为 X ≤ c₁ 或 X ≥ c₂。若观测值 x 落入其中任一区域，则拒绝 H₀。
Alternatively, some problems accept the p‑lue as 2 × min(P(X ≤ x), P(X ≥ x)). However, because the distribution is discrete, the two‑tail p‑value is not strictly symmetric, and the critical region method is usually preferred at A‑Level. Always follow the wording of the question.
另一种做法是取 p‑值 = 2 × min(P(X ≤ x), P(X ≥ x))。但由于离散分布并不严格对称，A‑Level 考试中通常更青睐临界区域法。请始终遵循题目的表述方式。
10. Step‑by‑Step Example (One‑Tailed Lower) | 分步示例（单尾下限检验）
A manufacturer claims that at least 40% 4. A gardener suspects the germination rate is lower and decides to test by planting seeds one by one until the first germination. The first germination occurs on the 9th seed. Test at the 5% significance level whether the true germination probability is less than 0.4.
某制造商声称其种子的发芽率至少为 40%，即 p = 0.4。一位园艺师怀疑发芽率更低，决定通过一颗一颗地播种直到第一颗发芽来进行检验。首次发芽发生在第 9 颗种子。在 5% 显著性水平下，检验真实发芽概率是否低于 0.4。
- Step 1: Define hypotheses. H₀: p = 0.4; H₁: p < 0.4.
- Step 2: Significance level α = 0.05.
- Step 3: Test statistic: X = number of seeds up to first germination. X ~ Geo(0.4) under H₀.
- Step 4: Observed value: x = 9.
- Step 5: p‑value = P(X ≥ 9) = (1 − 0.4)⁹⁻¹ = 0.6⁸ ≈ 0.0168.
- Step 6: Decision: 0.0168 < 0.05, so reject H₀.
- Step 7: Conclusion in context: There is sufficient evidence at the 5% level to suggest that the true germination rate is less than 40%.
中文翻译步骤：假设定义、显著性水平、统计量、观测值、p‑值计算、决策和结论。结果 p‑值 0.0168 < 0.05，拒绝 H₀，有足够证据表明真实发芽率低于 40%。
11. Common Pitfalls and Tips | 常见陷阱与技巧
common mistake is misidentifying which tail probability to use. Always check: if H₁ has a '<' sign, then larger x values are more extreme, so use P(X ≥ x); if H₁ has a '>' Another error is forgetting that the geometric distribution starts at x = 1, not 0. Double‑check the cumulative formulas: P(X ≤ x) = 1 − (1 − p)ˣ, not (1 − p)ˣ, and P(X ≥ x) = (1 − p)ˣ ritical values using logs, round conservatively in the appropriate direction (up for lower‑tail c, down for upper‑tail c) to ensure the probability condition is satisfied.
一个常见错误是弄错所使用的尾部概率。一定要检查：若 H₁ 含有 '<' 符号，则更大的 x 值更加极端，应使用 P(X ≥ x)；若 H₁ 含有 '>' 符号，则较小的 x 值为极端，应使用 P(X ≤ x)。另一个错误是忘记几何分布从 x = 1 开始，而非 0。务必再次核对累积公式：P(X ≤ x) = 1 − (1 − p)ˣ，而不是 (1 − p)ˣ；P(X ≥ x) = (1 − p)ˣ⁻¹。用对数求临界值时，要朝着正确的方向保守取整（下尾检验时向上取整，上尾检验时向下取整），以确保满足概率条件。
Also, remember that the hypothesis is always about the parameter p, not the observed statistic. State conclusions clearly in the context of the problem, e.g., 'There is evidence that the success probability has decreased', not 'The number of trials is too large'.
还要记住，假设总是针对参数 p，而非观测统计量本身。结论要清晰结合问题背景，例如“有证据表明成功概率下降了”，而不是“试验次数太大了”。
12. Exam‑Style Quick Reference | 考试速查表
| Scenario 情形 | p‑value | Critical Region 临界区域 |
|---|---|---|
| H₁: p < p₀ | P(X ≥ x) = (1 − p₀)ˣ⁻¹ | X ≥ c, where c = ⌈1 + ln α / ln(1 − p₀)⌉ |
| H₁: p > p₀ | P(X ≤ x) = 1 − (1 − p₀)ˣ | X ≤ c, where c = ⌊ln(1 − α) / ln(1 − p₀)⌋ |
This concise table summarises the two one‑ ling and floor symbols indicate rounding up or down to maintain the ≤ α condition. For two‑ th H₁: p ≠ p₀, apply the critical region method using α/2 on each side as described above.
这张简明表格总结了几何分布的两种单尾检验。天花板符号⌈⌉和地板符号⌊⌋分别表示向上或向下取整以维持 ≤ α 的条件。对于 H₁: p ≠ p₀ 的双尾检验，请按前文所述对每侧使用 α/2 的临界区域法。
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