离散随机变量的累积分布函数
累积分布函数（Cumulative Distribution Function）是概率论中的重要概念。如果随机变量 \(X\) 的某个特定值为 \(x\)，那么随机变量 \(X\) 小于或等于 \(x\) 的概率记为 \(F(x)\)。
The Cumulative Distribution Function (CDF) is an important concept in probability theory. If a particular value of the random variable \(X\) is \(x\), then the probability that the random variable \(X\) is less than or equal to \(x\) is denoted as \(F(x)\).
累积分布函数的定义
Definition of Cumulative Distribution Function
\[F(x) = P(X \leq x)\]
\(F(x)\) 是通过将所有小于或等于 \(x\) 的结果的概率相加得到的。
\(F(x)\) is found by adding together all the probabilities for those outcomes that are less than or equal to \(x\).
注意 / Note：
与概率分布类似，累积分布函数可以用表格形式表示。
Like a probability distribution, a cumulative distribution function can be written as a table.
让我们通过具体例子来学习如何构造累积分布函数。
Let's learn how to construct cumulative distribution functions through specific examples.
例 / Example：
掷两枚公平的硬币。随机变量 \(X\) 表示出现的正面朝上的硬币数量。
Two fair coins are tossed. The random variable \(X\) is the number of heads that appear on the two coins.
绘制一张表格显示 \(X\) 的累积分布函数。
Draw a table to show the cumulative distribution function for \(X\).
| 样本空间为 HH, HT, TH, TT | ||||
|---|---|---|---|---|
| 正面朝上数量 \(x\) | 0 | 1 | 2 | |
| \(P(X = x)\) | 0.25 | 0.5 | 0.25 | |
| \(F(x)\) | 0.25 | 0.75 | 1 | |
计算过程：
\(F(0) = P(0) = 0.25\)
\(F(1) = P(0) + P(1) = 0.25 + 0.5 = 0.75\)
\(F(2) = P(0) + P(1) + P(2) = 0.25 + 0.5 + 0.25 = 1\)
也可以使用 \(F(1) + P(2) = 0.75 + 0.25 = 1\)
解释 / Explanation：
这个例子展示了如何从概率分布构造累积分布函数。对于每个可能值 \(x\)，累积分布函数 \(F(x)\) 是所有小于或等于 \(x\) 的概率之和。
This example shows how to construct a cumulative distribution function from a probability distribution. For each possible value \(x\), the cumulative distribution function \(F(x)\) is the sum of all probabilities less than or equal to \(x\).
例 6.2.1 / Example 6.2.1：
离散随机变量 \(X\) 的累积分布函数 \(F(x)\) 定义为：
The discrete random variable \(X\) has a cumulative distribution function \(F(x)\) defined by:
\[F(x) = \frac{(x + k)}{8}, \quad x = 1, 2, 3\]
a) 求 \(k\) 的值。
b) 绘制累积分布函数的分布表。
c) 写出 \(F(2.6)\) 的值。
d) 求 \(X\) 的概率分布。
解答 / Solution：
a) \(F(3) = 1\)（因为所有值都小于或等于3）
所以 \(\frac{3 + k}{8} = 1\)
\(3 + k = 8\)
\(k = 5\)
b) \(F(2) = \frac{2 + 5}{8} = \frac{7}{8}\)
\(F(1) = \frac{1 + 5}{8} = \frac{6}{8} = \frac{3}{4}\)
| \(x\) | 1 | 2 | 3 |
|---|---|---|---|
| \(F(x)\) | \(\frac{3}{4}\) | \(\frac{7}{8}\) | 1 |
c) \(F(2.6) = F(2) = \frac{7}{8}\)
（因为 \(X\) 在2和3之间没有取值，所以 \(X \leq 2.6\) 等同于 \(X \leq 2\)）
d) \(P(X=1) = F(1) = \frac{3}{4}\)
\(P(X=2) = F(2) - F(1) = \frac{7}{8} - \frac{3}{4} = \frac{7}{8} - \frac{6}{8} = \frac{1}{8}\)
\(P(X=3) = F(3) - F(2) = 1 - \frac{7}{8} = \frac{1}{8}\)
重要提醒 / Important Note：
记住 \(X\) 是离散随机变量，所以不进行插值。\(F(2.6)\) 的含义是 \(P(X \leq 2.6)\)，但由于 \(X\) 在2和3之间没有取值，所以这等同于 \(P(X \leq 2)\)。
Remember that \(X\) is a discrete random variable so you do not interpolate. \(F(2.6)\) means \(P(X \leq 2.6)\), but since \(X\) doesn't take any values between 2 and 3, this is the same as \(P(X \leq 2)\).
累积分布函数的关键特征：
Key Features of Cumulative Distribution Function:
学习建议 / Learning Tips：
熟练掌握累积分布函数的构造方法对于理解概率分布的性质至关重要。多做练习有助于加深理解。
Mastering the construction method of cumulative distribution functions is crucial for understanding the properties of probability distributions. More practice helps deepen understanding.