Geometric Distribution Critical Regions for A-Level Further Maths | A-Level进阶数学：几何分布临界区域的求法

📚 Geometric Distribution Critical Regions for A-Level Further Maths | A-Level进阶数学：几何分布临界区域的求法

In A-Level Further Mathematics, the geometric distribution models the number of trials needed to achieve the first success. Finding critical regions for hypothesis tests involving this distribution is a key skill for Edexcel exam success.

在 A-Level 进阶数学中，几何分布用于描述首次成功所需试验次数。求解该分布假设检验中的临界区域，是 Edexcel 考试取得高分的关键技能。

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1. The Geometric Distribution Recall | 几何分布回顾

For a random variable X with geometric distribution X ∼ Geo(p), where p is the probability of success on each trial, the probability mass function is:

对于随机变量 X，若 X ∼ Geo(p)，其中 p 是每次试验成功的概率，则其概率质量函数为：

P(X = x) = (1−p)^x−1p, for x = 1, 2, 3, …

The cumulative distribution function used for finding critical regions is:

用于求临界区域的累积分布函数为：

P(X ≤ x) = 1 − (1−p)^x

A useful tail probability follows from the memoryless property:

由无记忆性可得一个有用的尾概率：

P(X ≥ x) = (1−p)^x−1

These three formulas are the foundation for every critical-region calculation.

这三个公式是所有临界区域计算的基础。

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2. Hypothesis Testing with a Geometric Distribution | 几何分布的假设检验

In a hypothesis test, we start with a null hypothesis H₀: p = p₀. The alternative hypothesis can be one-sided or two-sided, but for the geometric distribution critical regions are almost always one-sided.

在假设检验中，我们先给出原假设 H₀: p = p₀。备择假设可以是单侧或双侧，但几何分布临界区域几乎总是单侧的。

A critical region is a set of values of X for which we reject H₀. The total probability of observing a value in this region, calculated under H₀, must be less than or equal to the significance level α.

临界区域是使我们需要拒绝原假设 H₀ 的 X 取值集合。在原假设成立的条件下，落入该区域的概率之和必须小于等于显著性水平 α。

• The critical region is chosen so that it contains the most extreme values in the direction suggested by H₁.
• 临界区域应包含 H₁ 所暗示方向上的最极端取值。

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3. Deciding the Direction of the Rejection Region | 判断拒绝域方向

For the geometric distribution, small values of X mean success happens early, so p is likely large. Large values of X mean success is delayed, so p is likely small.

对于几何分布，较小的 X 表示较早成功，因此 p 可能较大；较大的 X 表示成功较晚，因此 p 可能较小。

• If H₁: p < p₀, then we expect X to be large. Use a right-tailed critical region {X ≥ c}.
• 若 H₁: p < p₀，则我们预期 X 较大，应使用右尾临界区域 {X ≥ c}。
• If H₁: p > p₀, then we expect X to be small. Use a left-tailed critical region {X ≤ c}.
• 若 H₁: p > p₀，则我们预期 X 较小，应使用左尾临界区域 {X ≤ c}。

Getting this direction wrong is the most common student error.

方向判断错误是学生最常见的错误。

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4. Right-Tailed Critical Value: Finding c | 右尾临界值：求 c

For H₁: p < p₀, reject H₀ when X is too large. The critical region is {X ≥ c}, where c is the smallest integer satisfying:

对于 H₁: p < p₀，当 X 过大时拒绝 H₀。临界区域为 {X ≥ c}，其中 c 是满足下式的最小整数：

P(X ≥ c) = (1−p₀)^c−1 ≤ α

Use logarithms to solve the inequality:

使用对数求解不等式：

c − 1 ≥ ^{ln α}⁄_{ln(1 − p0)}

Therefore c = 1 + lceil lnα / ln(1−p₀) rceil, where lceil · rceil is the ceiling function. Remember that ln(1−p₀) is negative, so the inequality sign reverses when dividing.

因此 c = 1 + lceil lnα / ln(1−p₀) rceil，其中 lceil · rceil 是向上取整。记住 ln(1−p₀) 为负数，因此相除时需要变号。

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5. Left-Tailed Critical Value: Finding c | 左尾临界值：求 c

For H₁: p > p₀, reject H₀ when X is too small. The critical region is {X ≤ c}, where c is the largest integer satisfying:

对于 H₁: p > p₀，当 X 过小时拒绝 H₀。临界区域为 {X ≤ c}，其中 c 是满足下式的最大整数：

P(X ≤ c) = 1 − (1−p₀)^c ≤ α

Rearrange to obtain:

整理得到：

c ≤ ^{ln(1 − α)}⁄_{ln(1 − p0)}

So c = lfloor ln(1−α) / ln(1−p₀) rfloor, where lfloor · rfloor is the floor function. If this value is less than 1, then no non-empty critical region exists at that significance level.

因此 c = lfloor ln(1−α) / ln(1−p₀) rfloor，其中 lfloor · rfloor 是向下取整。如果该值小于 1，则在该显著性水平下不存在非空临界区域。

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6. Worked Example: Right-Tailed Test | 实例：右尾检验

Let X ∼ Geo(p). Test H₀: p = 0.2 against H₁: p < 0.2 at the 5% significance level.

设 X ∼ Geo(p)。在 5% 显著性水平下检验 H₀: p = 0.2 对 H₁: p < 0.2。

We need the smallest c such that (0.8)^c−1 ≤ 0.05.

我们需要最小整数 c，使得 (0.8)^c−1 ≤ 0.05。

(c−1)ln(0.8) ≤ ln(0.05) ⇒ c−1 ≥ 13.43

Thus c−1 = 14 and c = 15. The critical region is {X ≥ 15}.

因此 c−1 = 14，c = 15。临界区域为 {X ≥ 15}。

The actual significance is P(X ≥ 15) = 0.8¹⁴ = 0.04398, which is below 0.05 as required.

实际显著性水平为 P(X ≥ 15) = 0.8¹⁴ = 0.04398，如要求所料低于 0.05。

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7. Worked Example: Left-Tailed Test | 实例：左尾检验

Suppose X ∼ Geo(p). Test H₀: p = 0.05 against H₁: p > 0.05 at the 5% significance level.

假设 X ∼ Geo(p)。在 5% 显著性水平下检验 H₀: p = 0.05 对 H₁: p > 0.05。

We want the largest c such that 1 − (0.95)^c ≤ 0.05.

我们需要最大整数 c，使得 1 − (0.95)^c ≤ 0.05。

(0.95)^c ≥ 0.95 ⇒ c ≤ ln 0.95ln 0.95 = 1

So c = 1. The critical region is {X ≤ 1}, with P(X ≤ 1) = 0.05 exactly.

所以 c = 1。临界区域为 {X ≤ 1}，且 P(X ≤ 1) = 0.05，恰好等于显著性水平。

If p₀ = 0.2 instead, the floor calculation would give c = 0, meaning that no critical region exists because P(X = 1) = 0.2 > 0.05.

若换作 p₀ = 0.2，向下取整会得到 c = 0，意味着不存在临界区域，因为 P(X = 1) = 0.2 > 0.05。

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8. Using Tables to Verify Critical Regions | 用概率表验证临界区域

In an exam, you may be given a cumulative probability table for specific p values. To find a critical region, scan the table for the first probability that falls below or crosses α.

考试中，你可能会获得针对特定 p 的累积概率表。为求临界区域，扫描表格找到第一个低于或超过 α 的概率。

x	1	2	3	…
P(X ≤ x)	p	1−(1−p)²	1−(1−p)³	…

For a right-tailed region, calculate P(X ≥ x) = 1 − P(X ≤ x−1) and locate where it first drops to α or below.

对于右尾区域，计算 P(X ≥ x) = 1 − P(X ≤ x−1)，找到首次降到 α 或更低的位置。

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9. Common Mistakes and Tips | 常见错误与技巧

• Mixing up the direction of the alternative hypothesis. Always ask: would a smaller or larger X support H₁?
• 弄错备择假设的方向。始终问自己：X 偏小还是偏大会支持 H₁？
• Forgetting that probabilities in a critical region must be ≤ α, not < α.
• 忘记临界区域概率必须满足 ≤ α，而不是 < α。
• Using P(X ≥ c) = 1 − (1−p)^c for the right tail, which is wrong. The correct expression is (1−p)^c−1.
• 右尾概率误用 P(X ≥ c) = 1 − (1−p)^c。正确表达式是 (1−p)^c−1。
• Not stating the actual significance level after finding the critical region. The exam reward requires this summarising step.
• 找到临界区域后不写出实际显著性水平。考试评分要求总结这一步。

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10. Summary | 总结

To find a geometric critical region, first decide the tail direction from H₁. Then use the appropriate tail formula and solve the inequality for c. Always verify that the final critical region has probability no greater than α.

求几何分布临界区域时，首先根据 H₁ 确定尾部方向，然后使用对应的尾概率公式求解不等式得到 c。务必验证最终临界区域的概率不超过 α。

With practice, these calculations become routine and secure full marks in the hypothesis-testing section of Edexcel Further Mathematics.

多加练习后，这些计算会变得非常熟练，从而在 Edexcel 进阶数学的假设检验部分获得满分。

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