离散随机变量
随机变量是一个变量，其值依赖于随机事件的结局。随机变量用大写字母表示，例如X或Y，而随机变量可能取的特定值用相应的小写字母表示，例如x或y。随机变量X取特定值x的概率记为P(X = x)。
A random variable is a variable whose value depends on the outcome of a random event. Random variables are written using upper case letters, for example X or Y. The particular values that the random variable can take are written using equivalent lower case letters, for example x or y. The probability that the random variable X takes a particular value x is written as P(X = x).
本节将详细介绍随机变量的基本概念，包括随机变量的定义、表示法、样本空间的概念，以及如何区分离散随机变量和连续随机变量。这些概念是学习概率统计学的基础。
This section will detail the basic concepts of random variables, including the definition of random variables, notation, the concept of sample space, and how to distinguish between discrete random variables and continuous random variables. These concepts form the foundation of probability and statistics.
随机变量是将随机实验的结果映射到实数的函数。大写字母表示随机变量，小写字母表示其可能取值。
A random variable is a function that maps the outcomes of a random experiment to real numbers. Uppercase letters denote random variables, lowercase letters denote their possible values.
随机变量可能取值的集合称为样本空间。随机变量的取值范围决定了其样本空间。
The range of values that a random variable can take is called its sample space. The sample space is determined by the possible values of the random variable.
如果变量只能取某些数值，则称其为离散随机变量。离散随机变量的样本空间是有限个或可列无限个值。
A variable is discrete if it can take only certain numerical values. The sample space of a discrete random variable consists of a finite number or countably infinite number of values.
变量是随机的，如果其结果在实验进行之前是未知的。变量是离散的，如果它只能取某些数值。
A variable is random if the outcome is not known until the experiment is carried out. A variable is discrete if it can take only certain numerical values.
\[P(X = x)\]
随机变量X取值为x的概率。
The probability that random variable X takes value x.
\[\sum P(X = x) = 1\]
随机变量所有可能取值的概率之和等于1。
The sum of probabilities for all possible values of a random variable equals 1.