Cumulative Probability Functions: A Step-by-Step Guide | 累积概率函数的求解方法

📚 Cumulative Probability Functions: A Step-by-Step Guide | 累积概率函数的求解方法

For IB Mathematics students, the cumulative distribution function (CDF) is one of the most powerful tools in probability. It converts a probability density function or a probability distribution table into a single function that gives the probability that a random variable is at most a given value. Mastering CDFs is essential for solving exam questions on normal distributions, binomial distributions, and continuous random variables.

对于 IB 数学学生来说，累积分布函数（CDF）是概率学中最强大的工具之一。它将概率密度函数或概率分布表转化为一个函数，用于给出随机变量不超过某个给定值的概率。掌握累积分布函数是解答正态分布、二项分布和连续随机变量考题的关键。

1. Definition of the Cumulative Distribution Function | 累积分布函数的定义

Let X be a random variable. The cumulative distribution function of X is defined as F(x) = P(X ≤ x) for all real x. It gives the total probability accumulated from the left up to the value x.

设 X 为一个随机变量。X 的累积分布函数定义为 F(x) = P(X ≤ x)，对所有实数 x 成立。它给出从左侧到 x 所累积的总概率。

Every CDF has three essential properties. First, 0 ≤ F(x) ≤ 1. Second, F is non-decreasing: if a ≤ b, then F(a) ≤ F(b). Third, the limit as x approaches −∞ is 0, and the limit as x approaches +∞ is 1.

每个累积分布函数都具有三个基本性质。第一，0 ≤ F(x) ≤ 1。第二，F 是非减函数：若 a ≤ b，则 F(a) ≤ F(b)。第三，当 x 趋于 −∞ 时极限为 0，当 x 趋于 +∞ 时极限为 1。

These properties provide a quick way to verify whether a given function is a valid CDF. In IB questions, you may be asked to decide whether a piecewise expression can represent a CDF; check monotonicity and the two limits first.

这些性质提供了一种快速检验给定函数是否为合法 CDF 的方法。在 IB 题目中，你可能会被要求判断某个分段表达式能否表示 CDF；此时应首先检查单调性和两个极限。

2. Discrete Random Variables: Building F(x) | 离散型随机变量：构建 F(x)

For a discrete random variable X with possible values x₁, x₂, … and probabilities p₁, p₂, …, the CDF is the cumulative sum of probabilities for all values less than or equal to x.

对于取值为 x₁, x₂, … 、概率为 p₁, p₂, … 的离散型随机变量 X，其累积分布函数是所有不大于 x 的取值的概率之和。

F(x) = Σ pᵢ, where the sum is over all i with xᵢ ≤ x

F(x) = Σ pᵢ，其中求和涵盖所有满足 xᵢ ≤ x 的 i

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