Cumulative Probabilities: Discrete, Binomial and Normal Distributions | 累积概率：离散分布、二项分布与正态分布

📚 Cumulative Probabilities: Discrete, Binomial and Normal Distributions | 累积概率：离散分布、二项分布与正态分布

In Edexcel A Level Mathematics, cumulative probabilities appear throughout statistical distributions: discrete random variables, the binomial distribution and the normal distribution. They answer questions such as “What is the probability of at most 5 successes?” or “What proportion of values lie below a given score?”. This article explains how cumulative probabilities are defined, calculated and inverted, with typical exam techniques.

在 Edexcel A Level 数学中，累积概率贯穿统计分布：离散随机变量、二项分布和正态分布。它们回答诸如“至多 5 次成功的概率是多少？”或“低于某个分数的值占多大比例？”之类的问题。本文解释累积概率的定义、计算和反向求解，并给出典型考试技巧。

1. What Is a Cumulative Probability? | 什么是累积概率？

For a random variable X, the cumulative probability F(x) is the probability that X takes a value less than or equal to x.

对于随机变量 X，累积概率 F(x) 表示 X 取值小于或等于 x 的概率。

F(x) = P(X ≤ x)

Because F(x) is a probability, it must always lie between 0 and 1. It is also a non-decreasing function: as x increases, F(x) cannot decrease.

由于 F(x) 是概率，它必须始终介于 0 和 1 之间。它也是非递减函数：随着 x 增大，F(x) 不会减小。

For discrete variables, F(x) is a step function; for continuous variables, F(x) is a smooth curve obtained by integrating the probability density function.

对于离散变量，F(x) 是阶梯函数；对于连续变量，F(x) 是通过对概率密度函数积分得到的光滑曲线。

2. Cumulative Probabilities for a Discrete Random Variable | 离散随机变量的累积概率

A discrete random variable has a probability mass function P(X = x). The cumulative probability F(x) is found by adding all probabilities up to x.

离散随机变量具有概率质量函数 P(X = x)。累积概率 F(x) 通过将 x 之前的所有概率相加得到。

F(x) = Σ P(X = r) for all r ≤ x

For example, if P(X=1)=0.1, P(X=2)=0.2, P(X=3)=0.3, P(X=4)=0.4, then F(3)=0.1+0.2+0.3=0.6.

例如，若 P(X=1)=0.1，P(X=2)=0.2，P(X=3)=0.3，P(X=4)=0.4，则 F(3)=0.1+0.2+0.3=0.6。

The cumulative table is useful for quick reading of “at most” probabilities.

累积表有助于快速读出“至多”概率。

3. Recovering Individual Probabilities from F(x) | 从 F(x) 还原单个概率

If you are given a cumulative distribution table, you can recover the individual probability P(X = x) by subtracting the previous cumulative probability.

如果给出累积分布表，你可以用当前累积概率减去前一个累积概率来还原单个概率 P(X = x)。

P(X = x) = F(x) − F(x − 1)

For a discrete variable, P(X = 3) = F(3) − F(2). In the example above, this is 0.6 − 0.

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