6.7 Using Discrete Uniform Distribution as a Model

6.7.1 核心概念总结 / Core Concepts Summary

离散均匀分布定义：

随机变量 \(X\) 在有限个离散值上取值，且每个值具有相同概率的分布。

Definition of Discrete Uniform Distribution:

A distribution where a random variable \(X\) takes values over a finite set of discrete values, with each value having the same probability.

\[P(X = x) = \frac{1}{n}\]

对于n个等可能的值。

离散均匀分布的特征：

Characteristics of Discrete Uniform Distribution:

等概率性：所有可能取值具有相同的概率。Equal Probability: All possible values have the same probability.
离散性：取值集合是有限的离散集合。Discreteness: The value set is a finite discrete set.
对称性：对于连续整数集合，分布是对称的。Symmetry: For consecutive integer sets, the distribution is symmetric.
标准化：可以通过线性变换标准化为标准形式。Standardization: Can be standardized to standard form through linear transformation.

6.7.2 标准公式总结 / Standard Formulas Summary

对于取值范围 \(\{1, 2, 3, \ldots, n\}\) 的离散均匀分布：

For discrete uniform distribution with values \(\{1, 2, 3, \ldots, n\}\):

\[\mathrm{E}(X) = \frac{n + 1}{2}\]

\[\operatorname{Var}(X) = \frac{(n + 1)(n - 1)}{12}\]

\[\sigma = \sqrt{\frac{(n + 1)(n - 1)}{12}}\]

骰子分布示例 / Dice Distribution Example：

对于六面骰子（n=6）：

\[\mathrm{E}(X) = \frac{6 + 1}{2} = 3.5\]

\[\operatorname{Var}(X) = \frac{(6 + 1)(6 - 1)}{12} = \frac{35}{12} \approx 2.917\]

公式适用性 / Formula Applicability：

这些公式仅适用于取值范围为1到n的连续整数集合。如果取值范围不同，需要使用一般方法计算。

These formulas only apply to consecutive integer sets ranging from 1 to n. If the value range is different, the general method needs to be used for calculation.

6.7.3 变换和扩展 / Transformations and Extensions

一般离散均匀分布：

对于取值范围 \(\{a, a+1, \ldots, b\}\)，可以通过线性变换 \(Y = X - (a-1)\) 转换为标准形式。

General Discrete Uniform Distribution:

For value range \(\{a, a+1, \ldots, b\}\), it can be converted to standard form through linear transformation \(Y = X - (a-1)\).

变换计算步骤：

Transformation Calculation Steps:

标准化：将原分布转换为 \(\{1, 2, \ldots, n\}\) 的形式。Standardize: Convert the original distribution to the form \(\{1, 2, \ldots, n\}\).
计算统计量：使用标准公式计算期望值和方差。Calculate statistics: Use standard formulas to calculate expected value and variance.
逆变换：将结果转换回原变量的统计量。Inverse transform: Transform the results back to the statistics of the original variable.

数字范围变换示例 / Number Range Transformation Example：

对于数字0-9的均匀分布，可以通过 \(X = R + 1\) 转换为1-10的分布，其中R ∈ {0,1,…,9}。

For a uniform distribution of numbers 0-9, it can be converted to a 1-10 distribution through \(X = R + 1\), where R ∈ {0,1,…,9}.

6.7.4 应用要点总结 / Application Key Points

建模原则 / Modeling Principles：

等可能性判断：只有当所有结果确实等可能时才使用离散均匀分布。
Equal probability judgment: Only use discrete uniform distribution when all outcomes are indeed equally likely.
有限集合：适用于有限个离散结果的情况。
Finite sets: Applicable to situations with finite discrete outcomes.
标准化技巧：利用线性变换将非标准形式转换为标准形式。
Standardization technique: Use linear transformations to convert non-standard forms to standard forms.
公式记忆：记住标准公式的形式，便于快速计算。
Formula memorization: Remember the form of standard formulas for quick calculation.

使用注意事项 / Usage Precautions：

确认所有结果确实等可能。
Confirm that all outcomes are indeed equally likely.
检查取值范围是否为连续整数。
Check if the value range is consecutive integers.
注意公式的适用前提条件。
Note the prerequisites for formula application.

6.7.5 思维导图总结 / Mind Map Summary

离散均匀分布知识体系：

核心概念	数学特征	计算公式	应用领域
• 等概率分布 • 有限离散集合 • 对称性分布	• 常数概率 • 整数取值 • 对称均值 • 可预测方差	• P(X=x) = 1/n • E(X) = (n+1)/2 • Var(X) = (n+1)(n-1)/12 • 线性变换反推	• 骰子游戏 • 彩票抽奖 • 随机选择 • 质量检验 • 公平竞争