离散随机变量的累积分布函数
本节将介绍离散随机变量的累积分布函数（Cumulative Distribution Function）。累积分布函数 \(F(x)\) 定义为随机变量 \(X\) 小于或等于特定值 \(x\) 的概率，即 \(F(x) = P(X \leq x)\)。
This section introduces the cumulative distribution function for discrete random variables. The cumulative distribution function \(F(x)\) is defined as the probability that the random variable \(X\) is less than or equal to a specific value \(x\), i.e., \(F(x) = P(X \leq x)\).
通过学习本节内容，您将掌握如何从概率分布构造累积分布函数，理解其表格表示方法，并学会如何从累积分布函数反推概率分布。这些概念是理解概率论和统计学的基础。
By studying this section, you will learn how to construct cumulative distribution functions from probability distributions, understand their tabular representation, and learn how to derive probability distributions from cumulative distribution functions. These concepts form the foundation for understanding probability theory and statistics.
对于随机变量X的某个值x，F(x)表示X小于或等于x的概率。通过将所有小于或等于x的结果概率相加构造而成。
For a particular value x of the random variable X, F(x) represents the probability that X is less than or equal to x. It is constructed by adding together all the probabilities for outcomes less than or equal to x.
累积分布函数可以用表格形式表示，显示每个可能值的累积概率，与概率分布表格类似。
The cumulative distribution function can be represented in tabular form, showing the cumulative probability for each possible value, similar to a probability distribution table.
概率分布和累积分布函数可以相互转换：累积分布函数是概率分布的累积和，概率分布是累积分布函数的差值。
Probability distributions and cumulative distribution functions can be converted into each other: the cumulative distribution function is the cumulative sum of the probability distribution, and the probability distribution is the difference of the cumulative distribution function.
\[F(x) = P(X \leq x)\]
\[F(x) = \sum_{k \leq x} P(X = k)\]
\[P(X = x) = F(x) - F(x-1)\]