📚 Binomial Cumulative Distribution Function | 二项累积分布函数
The binomial cumulative distribution function (CDF) is a fundamental tool in A-Level statistics. It allows you to calculate the probability of obtaining at most a certain number of successes in a fixed number of independent trials. In Edexcel examinations, understanding the binomial CDF is essential for solving real-world problems and interpreting probability statements accurately.
二项累积分布函数（CDF）是 A-Level 统计学中的一个基本工具。它可以帮助你计算在固定次数的独立试验中，获得至多某一数量成功次数的概率。在 Edexcel 考试中，理解二项累积分布函数对于解决实际问题以及准确解读概率表达式至关重要。
1. Introduction to Binomial Distribution | 二项分布简介
A binomial distribution arises when an experiment consists of a fixed number of trials, n, each with only two possible outcomes: success or failure. The probability of success, p, remains constant for each trial, and the trials are independent. The random variable X, representing the number of successes, follows a binomial distribution written as X ~ B(n, p).
当一个实验包含固定次数的试验 n，并且每次试验只有两种可能结果——成功或失败时，就会产生二项分布。每次试验的成功概率 p 保持不变，且各次试验相互独立。表示成功次数的随机变量 X 服从二项分布，记作 X ~ B(n, p)。
For example, if a fair coin is tossed 10 times and X is the number of heads, then X ~ B(10, 0.5). Each toss is independent and the probability of getting a head is always 0.5.
例如，如果抛掷一枚公平硬币 10 次，X 表示正面朝上的次数，则 X ~ B(10, 0.5)。每次抛掷相互独立，且得到正面的概率始终为 0.5。
2. Probability Mass Function vs Cumulative Distribution Function | 概率质量函数与累积分布函数
The probability mass function (PMF) gives the probability that the random variable X takes an exact value, denoted P(X = x). In contrast, the cumulative distribution function (CDF) gives the probability that X takes a value less than or equal to x, denoted P(X ≤ x). The CDF is obtained by summing the PMF values up to x.
概率质量函数（PMF）给出随机变量 X 取某个精确值的概率，记作 P(X = x)。相比之下，累积分布函数（CDF）给出 X 取值小于或等于 x 的概率，记作 P(X ≤ x)。CDF 是将 PMF 的值从 0 累加到 x 得到的。
Understanding the difference is important. For example, P(X = 3) is a single probability, while P(X ≤ 3) includes the probabilities for X = 0, 1, 2 and 3.
理解两者的区别很重要。例如，P(X = 3) 是单一取值的概率，而 P(X ≤ 3) 包含了 X = 0、1、2、3 的概率之和。
3. Defining Binomial CDF | 二项累积分布函数的定义
For a binomial random variable X ~ B(n, p), the cumulative distribution function is defined as the probability that X is at most x. In symbols, F(x) = P(X ≤ x). It can be expressed as a sum of individual binomial probabilities from k = 0 to k = x.
对于二项随机变量 X ~ B(n, p)，累积分布函数定义为 X 至多取值 x 的概率。用符号表示为 F(x) = P(X ≤ x)。它可以表示为从 k = 0 到 k = x 的各个二项概率之和。
The formula for the binomial CDF is:
F(x) = P(X ≤ x) = Σₖ₌₀ˣ ⁿCₖ pᵏ (1 − p)ⁿ⁻ᵏ
需要注意的是，这里 ⁿCₖ 是二项系数，等于 n! / [k!(n − k)!]。求和符号表示将 k 从 0 到 x 的所有项相加。
4. Formula and Summation Notation | 公式与求和符号
The binomial CDF uses summation notation to add up all relevant probabilities. The expression ⁿCₖ is read as ‘n choose k’ and counts the number of ways to choose k successes from n trials. The term pᵏ (1 − p)ⁿ⁻ᵏ gives the probability of any specific sequence with k successes and n − k failures.
二项累积分布函数使用求和符号将所有相关概率相加。表达式 ⁿCₖ 读作“n 选 k”，表示从 n 次试验中选出 k 次成功的方式数。项 pᵏ (1 − p)ⁿ⁻ᵏ 给出了在 k 次成功和 n − k 次失败条件下任意特定序列的概率。
When x = n, the CDF equals 1 because it includes all possible outcomes. When x = 0, the CDF simplifies to P(X = 0) = (1 − p)ⁿ.
当 x = n 时，累积分布函数等于 1，因为它包含了所有可能的结果。当 x = 0 时，累积分布函数简化为 P(X = 0) = (1 − p)ⁿ。
5. Using Statistical Tables | 使用统计表
In Edexcel examinations, you may be given binomial cumulative distribution tables. These tables list values of P(X ≤ x) for different values of n, p and x. To use the table, locate the correct n and p, then read across the row for the required x value.
在 Edexcel 考试中，你可能会得到二项累积分布表。这些表列出了不同 n、p 和 x 值下的 P(X ≤ x) 值。使用表格时，找到正确的 n 和 p，然后在对应行中查找所需的 x 值。
For example, if X ~ B(10, 0.3), to find P(X ≤ 4), locate the n = 10, p = 0.3 table, then find the entry in the row where x = 4. This entry is the cumulative probability.
例如，如果 X ~ B(10, 0.3)，要查找 P(X ≤ 4)，找到 n = 10，p = 0.3 的表格，然后找到 x = 4 所在行的条目。该条目就是累积概率。
If you need P(X < 5), remember that X is discrete, so P(X < 5) = P(X ≤ 4). This is a common conversion that must be handled carefully.
如果你需要计算 P(X < 5)，请记住 X 是离散变量，因此 P(X < 5) = P(X ≤ 4)。这是一个必须小心处理的常见转换。
6. Calculator Functions | 计算器功能
Most scientific calculators approved for Edexcel exams have built-in binomial functions. Typically, you can use the binomial CDF function, often labelled ‘BinomCD’ or ‘Binomial CDF’. You will need to enter the values for n, p and x, and the calculator returns P(X ≤ x).
大多数 Edexcel 考试允许使用的科学计算器都有内置的二项分布功能。通常，你可以使用二项累积分布函数，通常标记为 ‘BinomCD’ 或 ‘Binomial CDF’。你需要输入 n、p 和 x 的值，计算器会返回 P(X ≤ x)。
Some calculators may ask whether you want the lower or upper cumulative probability. For P(X ≤ x), choose the lower cumulative setting. For P(X ≥ x), you can either use the upper cumulative setting or apply the complement rule.
有些计算器可能会询问你想要下累积概率还是上累积概率。对于 P(X ≤ x)，选择下累积设置。对于 P(X ≥ x)，你可以使用上累积设置，或者应用补集规则。
7. Worked Example: Basic Calculation | 实例：基本计算
Suppose X ~ B(12, 0.4). Find P(X ≤ 5). Using the binomial CDF formula, we need the sum of probabilities for X = 0, 1, 2, 3, 4 and 5. From tables or calculator, we obtain approximately 0.6652.
假设 X ~ B(12, 0.4)，求 P(X ≤ 5)。使用二项累积分布函数公式，我们需要将 X = 0、1、2、3、4、5 的概率相加。查表或使用计算器，得到约为 0.6652。
This means there is about a 66.5% chance of observing 5 or fewer successes in 12 trials when the success probability is 0.4.
这意味着当成功概率为 0.4 时，在 12 次试验中观察到 5 次或更少成功的概率约为 66.5%。
8. Worked Example: Inequality Problems | 实例：不等式问题
Consider X ~ B(20, 0.25). Find P(4 ≤ X ≤ 8). This probability can be expressed as P(X ≤ 8) − P(X ≤ 3). We subtract the cumulative probability up to 3 because we want to exclude outcomes 0, 1, 2, 3.
考虑 X ~ B(20, 0.25)，求 P(4 ≤ X ≤ 8)。这个概率可以表示为 P(X ≤ 8) − P(X ≤ 3)。我们减去累积到 3 的概率，因为我们要排除 0、1、2、3 这些结果。
Using tables or calculator, P(X ≤ 8) ≈ 0.9591 and P(X ≤ 3) ≈ 0.2252. Therefore, P(4 ≤ X ≤ 8) ≈ 0.9591 − 0.2252 = 0.7339.
使用表格或计算器，P(X ≤ 8) ≈ 0.9591，P(X ≤ 3) ≈ 0.2252。因此，P(4 ≤ X ≤ 8) ≈ 0.9591 − 0.2252 = 0.7339。
This technique of subtracting two cumulative probabilities is extremely common in exam questions. Always ensure the lower bound is adjusted by subtracting P(X ≤ one less than the lower value).
这种减去两个累积概率的方法在考试题目中非常常见。务必确保下限调整正确，即减去 P(X ≤ 下限值减一)。
9. Relationship with Complement Rule | 与补集规则的关系
The complement rule states that the total probability is 1. Therefore, P(X ≥ x) = 1 − P(X ≤ x − 1). This is useful because tables often only provide cumulative probabilities for ‘less than or equal to’ a given value.
补集规则指出总概率为 1。因此，P(X ≥ x) = 1 − P(X ≤ x − 1)。这非常有用，因为表格通常只提供“小于或等于”某个值的累积概率。
For example, if X ~ B(15, 0.2) and you need P
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