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 | 几何分布参数 p 的假设检验

In Edexcel A Level Mathematics, hypothesis tests are not limited to binomial or normal models. The geometric distribution Geo(p) can also be tested, usually to decide whether the probability of success p has changed from a claimed value. Since the test statistic is the number of trials up to and including the first success, the critical region is built from geometric tail probabilities.

在Edexcel A Level数学中，假设检验不仅限于二项分布或正态分布。几何分布 Geo(p) 的参数 p 也可以进行检验，通常用于判断每次试验的成功概率 p 是否偏离了声称值。由于检验统计量是直到首次成功所需的试验次数，因此拒绝域需要由几何分布的尾部概率来确定。

1. Geometric Distribution Essentials | 几何分布基础

A discrete random variable X follows a geometric distribution, written X ~ Geo(p), if it counts the number of independent trials needed to obtain the first success. Each trial has probability p of success and 1 − p of failure.

如果离散随机变量 X 表示在独立重复试验中首次成功所需的试验次数，则称 X 服从几何分布，记为 X ~ Geo(p)。每次试验成功的概率为 p，失败的概率为 1 − p。

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

The key cumulative results used in hypothesis tests are the lower-tail and upper-tail probabilities:

假设检验中常用的累积概率是下尾概率和上尾概率：

P(X ≤ x) = 1 − (1 − p)^x

P(X ≥ x) = (1 − p)^{x − 1}

The second result follows because X ≥ x means the first x − 1 trials were all failures. These two formulas allow quick calculation of critical regions and p-values without geometric tables.

第二个公式成立的原因是 X ≥ x 表示前 x − 1 次试验全部失败。利用这两个公式可以快速计算拒绝域和 p 值，无需查几何分布表。

2. Hypotheses for p | 关于 p 的假设

The null hypothesis is always H₀: p = p₀, where p₀ is the claimed probability of success. The alternative hypothesis depends on the wording of the context or the suspicion being investigated.

原假设始终为 H₀: p = p₀，其中 p₀ 是声称的成功概率。备择假设则由题意或所怀疑的方向决定。

For a one-tailed test, use H₁: p < p₀ when there is evidence that the success probability has decreased. Use H₁: p > p₀ when there is evidence that the success probability has increased. For a two-tailed test, use H₁: p ≠ p₀ when the claim is simply that the probability has changed, with no direction specified.

若怀疑成功概率减小，则使用单尾检验 H₁: p < p₀。若怀疑成功概率增大，则使用 H₁: p > p₀。如果只关心概率是否发生变化而不指定方向，则使用双尾检验 H₁: p ≠ p₀。

3. Test Statistic and Its Distribution | 检验统计量及其分布

The natural test statistic is X, the observed number of trials until the first success. Under H₀, X ~ Geo(p₀), so all probabilities are calculated using p = p₀. There is no need to use a normal approximation in Edexcel geometric hypothesis tests.

自然使用的检验统计量是 X，即观察到首次成功所需的试验次数。在 H₀ 成立时，X ~ Geo(p₀)，因此所有概率都使用 p = p₀ 来计算。在 Edexcel

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）