Finding Critical Regions for a Poisson Distribution | 泊松分布的临界域确定

📚 Finding Critical Regions for a Poisson Distribution | 泊松分布的临界域确定

When you carry out a hypothesis test for a Poisson mean, the critical region tells you exactly which observed counts should lead you to reject H₀. This article shows how to find critical regions for one-tailed and two-tailed Poisson tests at a given significance level.

当你对泊松均值进行假设检验时，临界域会告诉你哪些观测计数会导致拒绝原假设 H₀。本文介绍如何在给定显著性水平下，为单尾和双尾泊松检验确定临界域。

1. Recap: The Poisson Distribution | 回顾：泊松分布

A Poisson distribution is used to model the number of times an event occurs in a fixed interval of time or space, when events happen independently and at a constant average rate. If X ~ Po(λ), then the mean and variance are both λ.

泊松分布用于对固定时间或空间区间内事件发生次数进行建模，条件是事件独立发生且平均发生率恒定。若 X ~ Po(λ)，则均值与方差均为 λ。

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

The formula uses e⁻λ as the exponential term, λˣ as λ raised to the power x, and x! as the factorial of x.

公式中 e⁻λ 是自然指数项，λˣ 表示 λ 的 x 次方，x! 是 x 的阶乘。

2. What Is a Critical Region? | 什么是临界域？

A critical region, also called a rejection region, is the set of values of the test statistic for which the null hypothesis is rejected. The boundary values are called critical values. If the observed test statistic falls in the critical region, the result is significant at the chosen level.

临界域，也称为拒绝域，是检验统计量的取值集合，当统计量落入该集合时拒绝原假设。边界值称为临界值。如果观测检验统计量落入临界域，则结果在该选定水平下显著。

In a Poisson hypothesis test, the test statistic is the observed count X. The critical region is therefore a range of counts such as X ≤ c, X ≥ c, or both tails together.

在泊松假设检验中，检验统计量是观测计数 X。因此临界域是一个计数范围，例如 X ≤ c、X ≥ c，或两个尾部同时出现。

3. Hypothesis Test Setup for Poisson | 泊松分布假设检验设置

For a Poisson test, the null hypothesis usually states that the population mean λ equals a known or claimed value λ₀. The alternative hypothesis determines whether the test is lower-tail, upper-tail, or two-tailed.

对于泊松检验，原假设通常表述为总体均值 λ 等于已知值或声称值 λ₀。备择假设决定检验是下限单尾、上限单尾还是双尾。