The Cumulative Distribution Function | 累积分布函数

📚 The Cumulative Distribution Function | 累积分布函数

The cumulative distribution function (CDF) is one of the most important tools for describing a continuous random variable in A-Level Mathematics. It gives the probability that a random variable X takes a value less than or equal to a given number x. Understanding the relationship between the CDF and the probability density function (PDF) is essential for Edexcel Statistics modules.

累积分布函数（CDF）是A-Level数学中描述连续随机变量的最重要工具之一。它给出随机变量X取值小于或等于给定数值x的概率。理解CDF与概率密度函数（PDF）之间的关系是Edexcel统计模块的核心内容。

1. Introduction to CDF | 累积分布函数简介

In probability theory, the cumulative distribution function F(x) of a continuous random variable X is defined as F(x) = P(X ≤ x). It accumulates all probability from the lower bound of the distribution up to the value x. Unlike the PDF, which gives the relative likelihood at a single point, the CDF gives the total probability up to that point.

在概率论中，连续随机变量X的累积分布函数F(x)定义为F(x) = P(X ≤ x)。它将分布下限到数值x之间的所有概率累积起来。与给出单点相对似然性的PDF不同，CDF给出到该点为止的总概率。

F(x) = P(X ≤ x)

For example, if X is the height of a randomly chosen student, then F(170 cm) is the probability that the student is at most 170 cm tall. This interpretation is central to exam questions.

例如，若X是随机选择的学生身高，那么F(170 cm)表示该学生身高不超过170 cm的概率。这种理解方式在考试问题中非常关键。

2. Formal Definition | 正式定义

For a continuous random variable X with probability density function f(x), the cumulative distribution function is given by the integral F(x) = ∫₋∞ˣ f(t) dt. Here t is a dummy variable of integration, and the lower limit is negative infinity because the total probability from the far left of the distribution is being accumulated.

对于概率密度函数为f(x)的连续随机变量X，累积分布函数由积分 F(x) = ∫₋∞ˣ f(t) dt 给出。这里t是积分虚拟变量，积分下限为负无穷，因为从分布最左端开始累积总概率。

F(x) = ∫₋∞ˣ f(t) dt

The CDF can also be defined for discrete random variables as a sum, but Edexcel A-Level mainly examines the continuous case. In the continuous case, F(x) is a smooth, non-decreasing function wherever the PDF is positive.

对于离散随机变量，CDF也可定义为求和形式，但Edexcel A-Level主要考查连续情况。在连续情况下，只要PDF为正，F(x)就是光滑且非递减的函数。

3. Key Properties | 关键性质

The CDF has several important properties. First, 0 ≤ F(x) ≤ 1 for all real x. Second, as x → −∞, F(x) → 0, because no probability has been accumulated. Third, as x → ∞, F(x) → 1, because all probability has been accumulated. Fourth, F(x) is a non-decreasing function: if a < b, then F(a) ≤ F(b).

CDF具有几个重要性质。第一，对所有实数x，0 ≤ F(x) ≤ 1。第二，当x → −∞时，F(x) → 0，因为还没有累积任何概率。第三，当x → ∞时，F(x) → 1，因为所有概率都已被累积。第四，F(x)是非递减函数：如果a < b，则F(a) ≤ F(b)。

These properties are often used in exam questions to determine unknown constants in a piecewise CDF or to check whether a given function can be a valid CDF.

这些性质常用于考试题中，以确定分段CDF中的未知常数，或检验给定函数是否为有效的CDF。

4. From PDF to CDF | 由概率密度函数求CDF

If you are given a PDF f(x) defined on an interval, you can find the CDF by integrating. For example, if f(x) = kx² for 0 ≤ x ≤ 2 and 0 otherwise, you first find k by ensuring the total area under the PDF is 1. Then, for 0 ≤ x ≤ 2, F(x) = ∫₀ˣ kt² dt. For x < 0, F(x) = 0; for x > 2, F(x) = 1.

如果给定一个定义在某个区间上的PDF f(x)，可以通过积分求得CDF。例如，若f(x) = kx²（0 ≤ x ≤ 2，其他情况为0），首先通过确保PDF下方面积为1求出k。然后，对于0 ≤ x ≤ 2，F(x) = ∫₀ˣ kt² dt。当x < 0时，F(x) = 0；当x > 2时，F(x) = 1。

F(x) = ∫₀ˣ kt² dt for 0 ≤ x ≤ 2

Remember to use a dummy variable such as t inside the integral, and to evaluate the definite integral with the lower limit equal to the start of the non-zero part of the PDF, not always negative infinity.

请记住在积分内使用虚拟变量（如t），并且将定积分的下限取为PDF非零部分的起点，而并不总是负无穷。

5. From CDF to PDF | 由CDF求PDF

The probability density function is the derivative of the cumulative distribution function: f(x) = F′(x). This relationship follows from the Fundamental Theorem of Calculus, provided F is differentiable at x. In exam problems, you may be given a piecewise CDF and asked to find the PDF by differentiation.

概率密度函数是累积分布函数的导数：f(x) = F′(x)。这一关系来自微积分基本定理，前提是F在x处可导。在考试问题中，可能会给一个分段CDF，要求通过微分求PDF。

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