📚 Cumulative Distribution Functions | 累积分布函数
In probability theory and statistics, the cumulative distribution function (CDF) of a random variable X is a fundamental tool that gives the probability that X will take a value less than or equal to x, for any real number x. It accumulates probabilities from the left up to a given point, providing a complete description of the distribution of X. Understanding the CDF is essential for solving problems involving intervals, medians, percentiles, and for linking probability density functions to probabilities in continuous settings.
在概率论与统计学中，随机变量X的累积分布函数（CDF）是一个基本工具，它给出了对于任意实数x，X取值小于或等于x的概率。它从左侧将概率累积到给定点，完整地描述了X的分布。理解累积分布函数对于解决涉及区间、中位数、百分位数的问题，以及在连续情形下将概率密度函数与概率联系起来至关重要。
1. Definition and Notation | 定义与符号
plied to discrete, continuous, or mixed random variables. The function F(x) is defined on the entire real line, and its value at x represents the total probability accumulated up to and including x. In many textbooks, the notation FX(x) is used to emphasize the random variable.
对于随机变量X，累积分布函数记为F(x)，定义为对所有实数x，F(x) = P(X ≤ x)。它适用于离散型、连续型或混合型随机变量。函数F(x)定义在整个实数轴上，其在x点的值表示直到x（包括x）为止累积的总概率。在许多教科书中，使用记号FX(x)以强调随机变量。
F(x) = P(X ≤ x), x ∈ ℝ
2. Properties of the CDF | 累积分布函数的性质
A cumulative distribution function F(x) has several key properties that make it a powerful tool in probability. Each property follows directly from the axioms of probability.
累积分布函数F(x)具有几个关键性质，使其成为概率论中的强大工具。每条性质都直接源于概率公理。
) ≤ F(x₂). Adding more possible values can only increase or keep the probability the same.
1. 非减性：若 x₁ ≤ x₂，则 F(x₁) ≤ F(x₂)。增加更多可能的取值只会增大或保持概率不变。
2. Limits at infinity: limx→−∞ F(x) = 0 and limx→∞ is less than a very small number approaches 0, and the total probability that X is less than a very large number approaches 1.
2. 无穷处的极限：limx→−∞ F(x) = 0；limx→∞ F(x) = 1。X小于极小数的概率趋于0，X小于极大数的概率趋于1。
ght-continuity: F(x) is right-continuous, meaning that for every real number a, F(a) = limx→a⁺ F(x). This property is particularly important when the random variable has point masses (discrete distributions).
3. 右连续：F(x)是右连续的，即对于每个实数a，F(a) = limx→a⁺ F(x)。当随机变量具有点概率（离散分布）时，这一性质尤为重要。
b) = F(b) − F(a). This is the most frequently used formula when working with a known CDF.
4. 区间概率：P(a < X ≤ b) = F(b) −
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