Cumulative Distribution Function (CDF)
Definition
English
The Cumulative Distribution Function (CDF) of a random variable is a function that gives the probability that will take a value less than or equal to . Mathematically, it is defined as:
中文
累积分布函数（CDF）是随机变量 的一个函数 ，它给出 取值小于或等于 的概率。数学上，它定义为：
Properties
English
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Non-decreasing: is a non-decreasing function.
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Limits:
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Right-continuous: is right-continuous, meaning .
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Probability Mass Function (PMF) Relation (for discrete random variables):
F_X(x) = \sum_{t \leq x} P(X = t)
F_X(x) = \int_{-\infty}^{x} f_X(t) , dt
### 中文 1. **非减性**：$F_X(x)$ 是一个非减函数。 2. **极限**： - $\lim_{x \to -\infty} F_X(x) = 0$ - $\lim_{x \to \infty} F_X(x) = 1$ 3. **右连续**：$F_X(x)$ 是右连续的，意味着 $\lim_{x \to x_0^+} F_X(x) = F_X(x_0)$。 4. **概率质量函数 ([[PMF&PDF#概率密度函数-pdf|PMF]]) 关系**（对于离散随机变量）：F_X(x) = \sum_{t \leq x} P(X = t)
5. **概率密度函数 ([[PMF&PDF#概率密度函数-pdf|PDF]]) 关系**（对于连续随机变量）：F_X(x) = \int_{-\infty}^{x} f_X(t) , dt
## Calculation Techniques ### English - **Discrete Random Variable**: To find the CDF of a discrete random variable, sum the probabilities of all outcomes less than or equal to $x$.F_X(x) = \sum_{t \leq x} P(X = t)
- **Continuous Random Variable**: To find the CDF of a continuous random variable, integrate the PDF from $-\infty$ to $x$.F_X(x) = \int_{-\infty}^{x} f_X(t) , dt
### 中文 - **离散随机变量**：为了找到离散随机变量的CDF，求所有小于或等于 $x$ 的结果的概率之和。 ==F_X(x) = \sum_{t \leq x} P(X = t)
- **连续随机变量**：为了找到连续随机变量的CDF，积分从 $-\infty$ 到 $x$ 的PDF。 ==F_X(x) = \int_{-\infty}^{x} f_X(t) , dt
## Important Points to Remember ### English - The CDF provides a complete description of the probability distribution of a random variable. - For continuous random variables, the derivative of the CDF with respect to $x$ gives the PDF:f_X(x) = \frac{d}{dx} F_X(x)
P(X = x) = F_X(x) - F_X(x-)
P(a < X \leq b) = F_X(b) - F_X(a)
### 中文 - CDF提供了随机变量的概率分布的完整描述。 - 对于连续随机变量，CDF关于 $x$ 的导数给出了PDF：f_X(x) = \frac{d}{dx} F_X(x)
P(X = x) = F_X(x) - F_X(x-)
P(a < X \leq b) = F_X(b) - F_X(a)
## Common Mistakes ### English - Confusing the CDF with the PDF. Remember, the CDF is the integral of the PDF for continuous variables and the sum of the PMF for discrete variables. - Forgetting that the CDF is right-continuous. - Misinterpreting the limits: Ensure you correctly understand that $F_X(x)$ approaches 0 as $x$ approaches $-\infty$ and 1 as $x$ approaches $\infty$. ### 中文 - 将CDF与PDF混淆。记住，对于连续变量，CDF是PDF的积分；对于离散变量，CDF是PMF的和。 - 忘记CDF是右连续的。 - 误解极限：确保你正确理解 $F_X(x)$ 当 $x$ 趋近 $-\infty$ 时趋近于0，当 $x$ 趋近 $\infty$ 时趋近于1。 ## Example ### English Given a discrete random variable $X$ with the following PMF:P(X = x) =
\begin{cases}
0.2 & \text{if } x = 1 \
0.5 & \text{if } x = 2 \
0.3 & \text{if } x = 3 \
0 & \text{otherwise}
\end{cases}
The CDF $F_X(x)$ is calculated as:F_X(x) =
\begin{cases}
0 & \text{if } x < 1 \
0.2 & \text{if } 1 \leq x < 2 \
0.7 & \text{if } 2 \leq x < 3 \
1 & \text{if } x \geq 3
\end{cases}
### 中文 给定一个离散随机变量 $X$，其PMF如下：P(X = x) =
\begin{cases}
0.2 & \text{如果 } x = 1 \
0.5 & \text{如果 } x = 2 \
0.3 & \text{如果 } x = 3 \
0 & 其他情况
\end{cases}
CDF $F_X(x)$ 计算如下：F_X(x) =
\begin{cases}
0 & \text{如果 } x < 1 \
0.2 & \text{如果 } 1 \leq x < 2 \
0.7 & \text{如果 } 2 \leq x < 3 \
1 & \text{如果 } x \geq 3
\end{cases}