摘要：概率论与数理统计是数学的一个重要分支，主要研究随机现象的数学性质和统计方法。在概率论中，随机变量是一个核心概念，它是一个将样本空间映射到实数集的函数。随机变量的数字特征是用来描述随机变量分布特性的数值，常用的数字特征包括以下几个：
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思维导图
Mind mapping
概率论与数理统计是数学的一个重要分支，主要研究随机现象的数学性质和统计方法。在概率论中，随机变量是一个核心概念，它是一个将样本空间映射到实数集的函数。随机变量的数字特征是用来描述随机变量分布特性的数值，常用的数字特征包括以下几个：
Probability theory and mathematical statistics is an important branch of mathematics, mainly studying the mathematical properties and statistical methods of random phenomena. In probability theory, a random variable is a core concept, which is a function that maps the sample space to the set of real numbers. The numerical characteristics of random variables are used to describe the distribution characteristics of random variables. The commonly used numerical characteristics include the following:
期望（数学期望、均值）：
Expectation (Mathematical Expectation, Mean):
定义：随机变量取值的加权平均，是概率论中最基本的数字特征之一。
Definition: The weighted average of the values taken by the random variable, which is one of the most basic numerical characteristics in probability theory.
Variance:
Definition: Describes the average degree of deviation between the values of a random variable and its expectation.
记作：Var(X)或D(X)。
Denoted as: Var(X) or D(X).
Standard Deviation:
定义：方差的正平方根，用于描述随机变量取值的离散程度。
Definition: The positive square root of the variance, used to describe the degree of dispersion of the values of a random variable.
记作：σ(X)，σ(X) = √Var(X)。
Denoted as: σ(X), σ(X) = √Var(X).
协方差：
Covariance:
定义：描述两个随机变量之间线性相关程度的量。
Definition: A measure that describes the degree of linear correlation between two random variables.
记作：Cov(X, Y)，Cov(X, Y) = E[(X - E(X))(Y - E(Y))]。
Denoted as: Cov(X, Y), Cov(X, Y) = E[(X - E(X))(Y - E(Y))].
定义：标准化后的协方差，用来衡量两个随机变量之间的线性相关程度。
Definition: The standardized covariance, used to measure the degree of linear correlation between two random variables.
记作：ρ(X, Y)，ρ(X, Y) = Cov(X, Y) / (σ(X)σ(Y))。
Denoted as: ρ(X, Y), ρ(X, Y) = Cov(X, Y) / (σ(X)σ(Y)).
矩：
Moments:
定义：随机变量的幂的期望值。
Definition: The expected value of the power of a random variable.
记作：对于随机变量X的第k阶原点矩为E(X^k)，第k阶中心矩为E[(X - E(X))^k]。
Denoted as: For the k-th order raw moment of random variable X as E(X^k), and the k-th order central moment as E[(X - E(X))^k].
这些数字特征为我们提供了随机变量的重要信息，有助于我们更好地理解随机现象的性质，并在实际应用中进行预测和分析。在数理统计中，这些特征也常被用于估计总体的参数，以及进行假设检验等。
These numerical characteristics provide important information about random variables, helping us to better understand the nature of random phenomena and to make predictions and analyses in practical applications. In mathematical statistics, these characteristics are also commonly used to estimate population parameters and conduct hypothesis testing, among other things.
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