📚 Testing a Discrete Uniform Distribution as a Model | 检验离散均匀分布作为模型
In AS-level statistics, a discrete uniform distribution is one of the simplest probability models: every possible outcome has exactly the same probability. When real data are collected, the observed frequencies will rarely match the expected frequencies perfectly, so we need a formal test to decide whether the differences are just random variation or evidence against the uniform model.
在 AS 阶段统计学中，离散均匀分布是最简单的概率模型之一：每个可能结果出现的概率完全相同。当我们收集真实数据时，观测频数很少会与期望频数完全一致，因此需要一个正式的检验来判断这些差异只是随机波动，还是说明均匀模型不成立。
1. What is a Discrete Uniform Distribution? | 什么是离散均匀分布？
A discrete random variable X has a discrete uniform distribution over k possible values if every value x₁, x₂, ..., xₖ is equally likely. The probability of each value is 1/k.
如果离散随机变量 X 在 k 个可能取值上服从离散均匀分布，那么每个取值 x₁、x₂、……、xₖ 出现的可能性相同，每个值的概率都是 1/k。
P(X = xᵢ) = 1/k, for i = 1, 2, ..., k
Common examples include rolling a fair six-sided die, selecting one digit from 0 to 9 at random, or spinning a fair spinner with equal sectors.
常见例子包括掷一个公平的六面骰子、从 0 到 9 中随机选一个数字，或者转动一个每个扇区大小相同的公平转盘。
2. Why Do We Test a Uniform Model? | 为什么要检验均匀模型？
Observed frequencies often deviate from the expected frequencies because of sampling variability. A goodness-of-fit test allows us to assess whether the observed deviations are too large to be explained by chance alone.
由于抽样波动，观测频数通常会偏离期望频数。拟合优度检验可以帮助我们判断观测偏差是否太大，以至于不能仅用随机性来解释。
For a discrete uniform model, the test compares the observed counts in each category with the counts we would expect if all categories were equally likely.
对于离散均匀模型，该检验将每个类别的观测频数与在所有类别等可能情况下所应得到的期望频数进行比较。
3. Setting Up the Hypotheses | 建立假设
The null hypothesis states that the data follow a discrete uniform distribution. The alternative hypothesis states that the data do not follow that uniform distribution.
原假设表示数据服从离散均匀分布；备择假设表示数据不服从该均匀分布。
- H₀: p₁ = p₂ = ... = pₖ = 1/k, so each category has the same probability.
- H₁: at least one pᵢ is not equal to 1/k.
In words: H₀ says the model is a discrete uniform distribution; H₁ says the model is not a discrete uniform distribution.
用文字来说：H₀ 表示模型是离散均匀分布；H₁ 表示模型不是离散均匀分布。
A chi-squared goodness-of-fit test is always an upper-tail test, because only large differences between observed and expected counts suggest a poor fit.
卡方拟合优度检验始终是上尾检验，因为只有观测频数与期望频数之间的差异很大时，才表明拟合效果差。
4. Observed and Expected Frequencies | 观测频数与期望频数
Suppose there are n observations in total and k categories. Under the null hypothesis of a discrete uniform distribution, the expected frequency for each category is:
假设总共有 n 个观测值，分为 k 个类别。在离散均匀分布的原假设下，每个类别的期望频数为：
Eᵢ = n / k, for every i
Observed frequencies Oᵢ are the actual counts recorded in the sample. If the uniform model is correct, Oᵢ should be reasonably close to n/k for each category.
观测频数 Oᵢ 是样本中实际记录的次数。如果均匀模型正确，每个类别的 Oᵢ 都应合理地接近 n/k。
5. The Chi-Squared Test Statistic | 卡方检验统计量
The test statistic measures how far the observed frequencies are from the expected frequencies. It is calculated as:
检验统计量用于衡量观测频数与期望频数之间的差距，计算公式为：
χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ
Here Oᵢ is the observed frequency, Eᵢ is the expected frequency, and the sum is taken over all k categories. A larger χ² value indicates a bigger difference between the data and the uniform model.
其中 Oᵢ 是观测频数，Eᵢ 是期望频数，求和覆盖所有 k 个类别。χ² 值越大，说明数据与均匀模型之间的差异越大。
6. Degrees of Freedom for a Uniform Model | 均匀模型的自由度
For a discrete uniform distribution with k categories and no parameters estimated from the data, the number of degrees of freedom is:
对于有 k 个类别且没有从数据中估计参数的离散均匀分布，自由度数为：
df = k − 1
We lose one degree of freedom because the total sample size n is fixed, so the last frequency is determined once the other k − 1 frequencies are known.
我们失去一个自由度，因为总样本量 n 是固定的，一旦知道前 k − 1 个频数，最后一个频数也随之确定。
7. Critical Values and Significance Level | 临界值与显著性水平
To make a decision, compare the calculated χ² value with the critical value from the chi-squared distribution at the chosen significance level, usually 5% or 1%.
作决策时，需要将计算出的 χ² 值与所选显著性水平（通常为 5% 或 1%）下卡方分布的临界值进行比较。
If χ² is greater than the critical value, there is sufficient evidence to reject H₀. If χ² is less than or equal to the critical value, we do not reject H₀.
如果 χ² 大于临界值，则有充分证据拒绝 H₀；如果 χ² 小于或等于临界值，则不拒绝 H₀。
For example, with df = 5 at the 5% level, the critical value is 11.07. With df = 9 at the 5% level, the critical value is 16.92.
例如，自由度为 5 时，5% 显著性水平下的临界值是 11.07；自由度为 9 时，5% 显著性水平下的临界值是 16.92。
8. Conditions for the Chi-Squared Test | 卡方检验的条件
The chi-squared goodness-of-fit test requires that the expected frequency in each category is sufficiently large. A common rule is that every expected frequency should be at least 5.
卡方拟合优度检验要求每个类别的期望频数足够大。常见规则是每个期望频数应至少为 5。
If some expected frequencies are less than 5, adjacent categories can be combined, or a larger sample should be collected. For a uniform model, this means n/k should normally be at least 5.
如果某些期望频数小于 5，可以合并相邻类别，或者收集更大的样本。
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