林深见鹿（十一）：概率论与数理统计（10）

摘要：Today, the editor brings the "Deep in the Woods, the Deer Appears (Part 11): Probability Theory and Mathematical Statistics (10)".

分享兴趣，传播快乐，增长见闻，留下美好！

亲爱的您，这里是LearningYard新学苑。

今天小编为大家带来

“林深见鹿（十一）：概率论与数理统计（10）”。

欢迎您的访问！

Share interest, spread happiness, increase knowledge, and leave beautiful.

Dear, this is the LearingYard New Academy!

Today, the editor brings the "Deep in the Woods, the Deer Appears (Part 11): Probability Theory and Mathematical Statistics (10)".

Welcome to visit!

今天来讲讲随机变量的数字特征。

Today, let's talk about the numerical characteristics of random variables.

数学期望是随机变量取值的加权平均，反映了随机变量可能的“平均水平”或“中心位置”。对于离散型随机变量，数学期望是每个可能取值与其概率的乘积之和；对于连续型随机变量，则是通过积分计算。数学期望具有线性性质，在实际应用中常用于预测和决策。

Themathematical expectation is the weighted average of the values of a random variable, reflecting its possible "average level" or "central position." For discrete random variables, the mathematical expectation is the sum of each possible value multiplied by its probability; for continuous random variables, it is calculated through integration. The mathematical expectation has the property of linearity and is often used in practical applications for prediction and decision-making.

方差则衡量随机变量取值与其数学期望的偏离程度，即数据的离散程度。方差越小，说明随机变量的取值越集中在期望附近；方差越大，则说明取值越分散。方差的算术平方根称为标准差，与随机变量具有相同的量纲，更便于实际解释。

Variance, on the other hand, measures the degree of deviation of a random variable's values from its mathematical expectation, indicating the dispersion of the data. The smaller the variance, the more concentrated the values of the random variable are around the expectation; the larger the variance, the more dispersed the values are. The square root of the variance is called the standard deviation, which has the same units as the random variable, making it more interpretable in practice.

协方差和相关系数用于描述两个随机变量之间的线性关系。协方差反映了两个变量的变化趋势是否一致，但其数值受变量量纲的影响。为消除量纲影响，引入了相关系数，它是一个标准化后的协方差，取值在-1到1之间，能够更清晰地刻画变量间的线性相关强度与方向。

Covariance and the correlation coefficient are used to describe the linear relationship between two random variables. Covariance reflects whether the trends of the two variables are consistent, but its value is influenced by the units of the variables. To eliminate the effect of units, the correlation coefficient is introduced. It is a standardized version of covariance, ranging from -1 to 1, and can more clearly describe the strength and direction of the linear relationship between variables.

矩是另一种描述随机变量分布特征的数学工具，包括原点矩和中心矩。数学期望是一阶原点矩，方差是二阶中心矩。高阶矩如偏度和峰度，还能进一步描述分布的形状特征。协方差矩阵则是多个随机变量之间协方差的矩阵表示，广泛应用于多元统计分析、主成分分析等领域，能够全面反映多个变量之间的相关结构。

Moments are another mathematical tool for describing the distribution characteristics of random variables, including origin moments and central moments. The mathematical expectation is the first-order origin moment, and the variance is the second-order central moment. Higher-order moments, such as skewness and kurtosis, can further describe the shape characteristics of the distribution. The covariance matrix is a matrix representation of the covariances between multiple random variables, widely used in multivariate statistical analysis, principal component analysis, and other fields, as it comprehensively reflects the correlation structure among multiple variables.

今天的分享就到这里了，

如果您对文章有独特的想法，

欢迎给我们留言。

让我们相约明天，

祝您今天过得开心快乐！

That's all for today's sharing.

If you have a unique idea about the article,

please leave us a message,

and let us meet tomorrow.

I wish you a nice day!

翻译：文心一言

参考资料：百度百科