离散随机变量的方差
本节将介绍离散随机变量的方差（Variance）。方差是测量随机变量与其期望值偏离程度的统计量，它描述了随机变量分布的散布程度。
This section introduces the variance of discrete random variables. Variance is a statistical measure of the degree of deviation of a random variable from its expected value, describing the spread of the random variable's distribution.
通过学习本节内容，您将掌握方差的定义、计算方法及其在概率统计中的应用。方差是理解随机变量分布特征的重要工具，在风险评估、质量控制等领域有着广泛的应用。
By studying this section, you will master the definition of variance, calculation methods, and its applications in probability and statistics. Variance is an important tool for understanding the characteristics of random variable distributions, with wide applications in risk assessment, quality control, and other fields.
方差测量随机变量与其期望值之间的平均平方偏差，是描述分布散布程度的重要统计量。
Variance measures the average squared deviation between a random variable and its expected value, and is an important statistic for describing the degree of dispersion of a distribution.
方差有两种等价的计算公式：定义公式和简便计算公式，后者通常更高效。
Variance has two equivalent calculation formulas: the definition formula and the convenient calculation formula, with the latter usually being more efficient.
方差越大，表示随机变量的取值越分散，越可能取与期望值差异大的值。
The larger the variance, the more dispersed the values of the random variable, and the more likely it is to take values that differ significantly from the expected value.
\[\operatorname{Var}(X) = \mathrm{E}[(X - \mathrm{E}(X))^2]\]
\[\operatorname{Var}(X) = \mathrm{E}(X^2) - [\mathrm{E}(X)]^2\]
\[\sigma = \sqrt{\operatorname{Var}(X)}\]