使用离散均匀分布作为模型
本节将介绍离散均匀分布（Discrete Uniform Distribution）及其在概率建模中的应用。当某些实验的结果具有等可能性时，离散均匀分布是合适的概率模型。
This section introduces the discrete uniform distribution and its applications in probability modeling. When the outcomes of certain experiments have equal probability, the discrete uniform distribution is an appropriate probability model.
通过学习本节内容，您将掌握离散均匀分布的定义、性质、标准公式及其在实际问题中的应用。这些知识对于理解等可能性事件的概率计算至关重要。
By studying this section, you will master the definition, properties, standard formulas of discrete uniform distributions, and their applications in practical problems. This knowledge is crucial for understanding probability calculations for equally likely events.
随机变量在有限个离散值上取值，且每个值具有相同概率的分布称为离散均匀分布。
A discrete uniform distribution is a distribution where a random variable takes values over a finite set of discrete values, with each value having the same probability.
所有可能的结果都具有相同的发生概率，这是离散均匀分布的核心特征。
All possible outcomes have the same probability of occurrence, which is the core characteristic of discrete uniform distributions.
对于取值范围为1到n的离散均匀分布，有固定的期望值和方差计算公式。
For discrete uniform distributions with values ranging from 1 to n, there are fixed formulas for calculating expected value and variance.
\[P(X = x) = \frac{1}{n}\]
对于n个等可能的值
\[\mathrm{E}(X) = \frac{n + 1}{2}\]
对于X ∈ {1, 2, ..., n}
\[\operatorname{Var}(X) = \frac{(n + 1)(n - 1)}{12}\]
对于X ∈ {1, 2, ..., n}