📚 Cumulative Distribution Function of the Binomial Distribution | 二项分布的累积分布函数
The binomial distribution is one of the most frequently used discrete probability distributions in A-Level Further Mathematics. While the probability mass function (PMF) gives the probability of exactly k successes, the cumulative distribution function (CDF) gives the probability of at most k successes. Understanding the CDF is essential for hypothesis testing, confidence intervals, and solving exam questions efficiently.
二项分布是 A-Level 进阶数学中最常用的离散型概率分布之一。概率质量函数（PMF）给出恰好 k 次成功的概率，而累积分布函数（CDF）则给出至多 k 次成功的概率。理解 CDF 对于假设检验、置信区间以及高效解答考试题目至关重要。
1. Definition of the Binomial Distribution | 二项分布的定义
Suppose a random variable X follows a binomial distribution with parameters n (number of trials) and p (probability of success). We write X ~ B(n, p). The probability of exactly r successes in n independent trials is given by the probability mass function:
假设随机变量 X 服从参数为 n（试验次数）和 p（成功概率）的二项分布。记作 X ~ B(n, p)。在 n 次独立试验中恰好出现 r 次成功的概率由概率质量函数给出：
P(X = r) = C(n, r) × pʳ × (1 − p)ⁿ⁻ʳ, r = 0, 1, 2, …, n
where C(n, r) = n! / (r! × (n − r)!) is the binomial coefficient. The distribution requires: (1) a fixed number of trials n; (2) each trial is independent; (3) only two outcomes per trial (success/failure); and (4) the probability p is constant across all trials.
其中 C(n, r) = n! / (r! × (n − r)!) 为二项式系数。该分布需要满足以下条件：（1）试验次数 n 固定；（2）每次试验相互独立；（3）每次试验只有两种结果（成功/失败）；（4）所有试验中成功概率 p 保持不变。
2. Definition of the Cumulative Distribution Function | 累积分布函数的定义
The cumulative distribution function of a discrete random variable X is defined as the probability that X takes a value less than or equal to a given value x. For a binomial distribution, the CDF is expressed as:
离散型随机变量 X 的累积分布函数定义为 X 取值小于或等于给定值 x 的概率。对于二项分布，CDF 表示为：
F(x) = P(X ≤ x) = Σᵣ₌₀ˣ C(n, r) × pʳ × (1 − p)ⁿ⁻ʳ
The cumulative sum runs from r = 0 to r = x, where x is an integer between 0 and n. For any x less than 0, F(x) = 0; for any x greater than or equal to n, F(x) = 1.
累积求和从 r = 0 到 r = x，其中 x 为 0 到 n 之间的整数。对于任何 x < 0，F(x) = 0；对于任何 x ≥ n，F(x) = 1。
3. Relationship Between PMF and CDF | 概率质量函数与累积分布函数的关系
There is a fundamental connection between the probability mass function P(X = r) and the cumulative distribution function F(x):
概率质量函数 P(X = r) 与累积分布函数 F(x) 之间存在基本关系：
F(x) = P(X ≤ x) = P(X = 0) + P(X = 1) + … + P(X = x)
P(X = x) = F(x) − F(x − 1)
The second expression is particularly useful when you have a table of cumulative probabilities. To find the probability of exactly x successes, simply subtract the cumulative probability at x − 1 from the cumulative probability at x.
第二个表达式在使用累积概率表时特别有用。要求恰好 x 次成功的概率，只需用 x 处的累积概率减去 x − 1 处的累积概率即可。
4. Complementary Probability: P(X > x) | 互补概率：P(X > x)
In many exam questions, you are asked for the probability that X exceeds a certain value. Since the total probability is 1, the complementary relationship is:
许多考试题目要求 X 超过某个值的概率。由于总概率为 1，互补关系为：
P(X > x) = 1 − P(X ≤ x) = 1 − F(x)
For example, if X ~ B(10, 0.4), then P(X > 6) = 1 − P(X ≤ 6). Similarly, P(X ≥ x) = 1 − P(X ≤ x − 1). These transformations allow you to use standard cumulative tables even when the question asks for upper-tail probabilities.
例如，若 X ~ B(10, 0.4)，则 P(X > 6) = 1 − P(X ≤ 6)。类似地，P(X ≥ x) = 1 − P(X ≤ x − 1)。这些变换使您即使面对上尾概率问题也能使用标准累积表。
5. Intervals and CDF | 区间概率与累积分布函数
The CDF can be used to calculate the probability that X lies within a specific interval. For integers a and b with 0 ≤ a ≤ b ≤ n:
CDF 可用于计算 X 落在特定区间内的概率。对于整数 a 和 b（0 ≤ a ≤ b ≤ n）：
P(a ≤ X ≤ b) = F(b) − F(a − 1)
P(a < X < b) = F(b − 1) − F(a)
The first formula includes both endpoints, while the second excludes them. Careful interpretation of inequalities is crucial. For example, P(X < 5) = P(X ≤ 4) = F(4), and P(X ≥ 5) = 1 − P(X ≤ 4) = 1 − F(4).
第一个公式包含两个端点，第二个公式则排除端点。仔细理解不等号至关重要。例如，P(X < 5) = P(X ≤ 4) = F(4)，而 P(X ≥ 5) = 1 − P(X ≤ 4) = 1 − F(4)。
6. Using Binomial Cumulative Distribution Tables | 使用二项分布累积表
In the Edexcel formula booklet, binomial cumulative probability tables are provided for various n and p values. These tables typically list P(X ≤ x) for x = 0, 1, …, n. To use them correctly, identify the row corresponding to your n value, the column corresponding to your p value, and then read the value at the appropriate x.
在 Edexcel 公式手册中，提供了不同 n 和 p 值的二项分布累积概率表。这些表通常列出 P(X ≤ x)，其中 x = 0, 1, …, n。要正确使用表格，请先找到与您的 n 值对应的行，再找到与您的 p 值对应的列，然后在相应的 x 处读取数值。
| x | P(X ≤ x) for n = 8, p = 0.35 |
| 0 | 0.0319 |
| 1 | 0.1691 |
| 2 | 0.4278 |
| 3 | 0.7064 |
| 4 | 0.8939 |
| 5 | 0.9747 |
The table above shows F(x) = P(X ≤ x) for X ~ B(8, 0.35). Notice that F(8) = 1 since the cumulative probability covers all possible outcomes. Always verify that you are reading P(X ≤ x) and not P(X = x).
上表显示了 X ~ B(8, 0.35) 时的 F(x) = P(X ≤ x)。注意 F(8) = 1，因为累积概率覆盖了所有可能的结果。务必确认您读取的是 P(X ≤ x) 而非 P(X = x)。
7. Using Calculators for the Binomial CDF | 使用计算器计算二项分布 CDF
Modern scientific and graphical calculators have built-in functions for binomial cumulative probabilities. On most models, you will find the function binomcdf(n, p, x) or an equivalent menu option. This function returns P(X ≤ x) directly, saving time and reducing arithmetic errors.
现代科学计算器和图形计算器内置了二项分布累积概率功能。在大多数型号上，您可以找到 binomcdf(n, p, x) 函数或等效的菜单选项。该函数直接返回 P(X ≤ x)，节省时间并减少算术错误。
For exact probabilities, use binompdf(n, p, x) on Casio models or the equivalent on other brands. To find P(a ≤ X ≤ b), compute binomcdf(n, p, b) − binomcdf(n, p, a − 1).
对于恰好概率，在 Casio 型号上使用 binompdf(n, p, x)，其他品牌也有等效功能。要计算 P(a ≤ X ≤ b)，请计算 binomcdf(n, p, b) − binomcdf(n, p, a − 1)。
8. Worked Example 1 | 例题 1
Let X ~ B(12, 0.6). Find P(X ≤ 8) and P(X > 5).
设 X ~ B(12, 0.6)。求 P(X ≤ 8) 和 P(X > 5)。
Solution: Using the cumulative table or calculator for n = 12, p = 0.6:
解答：使用 n = 12、p = 0.6 的累积表或计算器：
P(X ≤ 8) = F(8) = 0.7747
For the second part, P(X > 5) = 1 − P(X ≤ 5) = 1 − F(5). From the table, F(5) = 0.1582. Therefore:
对于第二部分，P(X > 5) = 1 − P(X ≤ 5) = 1 − F(5)。查表得 F(5) = 0.1582。因此：
P(X > 5) = 1 − 0.1582 = 0.8418
These results indicate that it is highly likely that more than 5 successes occur, while the probability of at most 8 successes is approximately 0.775.
这些结果表明发生超过 5 次成功的概率很高，而至多 8 次成功的概率约为 0.775。
9. Worked Example 2 | 例题 2
A fair die is rolled 10 times. Let X be the number of times a 6 appears. Find P(2 ≤ X ≤ 4).
一枚均匀骰子掷 10 次。设 X 为出现 6 的次数。求 P(2 ≤ X ≤ 4)。
Here, p = 1/6 ≈ 0.1667 and n = 10, so X ~ B(10, 1/6). We require:
此处 p = 1/6 ≈ 0.1667，n = 10，所以 X ~ B(10, 1/6)。我们需要：
P(2 ≤ X ≤ 4) = F(4) − F(1)
Using cumulative probabilities: F(4) = P(X ≤ 4) ≈ 0.9845 and F(1) = P(X ≤ 1) ≈ 0.4845. Therefore:
使用累积概率：F(4) = P(X ≤ 4) ≈ 0.9845，F(1) = P(X ≤ 1) ≈ 0.4845。因此：
P(2 ≤ X ≤ 4) = 0.9845 − 0.4845 = 0.5000
There is a 50% chance that the number of sixes rolled lies between 2 and 4 inclusive. This demonstrates how the CDF elegantly handles interval probabilities in binomial contexts.
掷出的 6 的次数在 2 到 4（包含端点）之间的概率为 50%。这展示了 CDF 如何优雅地处理二项分布中的区间概率问题。
10. Critical Values and Hypothesis Testing | 临界值与假设检验
In hypothesis testing with a binomial distribution, the CDF is used to find critical regions. For a one-tailed test at significance level α, the critical value c satisfies P(X ≤ c) ≤ α for a lower-tail test, or P(X ≥ c) ≤ α for an upper-tail test.
在使用二项分布的假设检验中，CDF 用于寻找临界区域。对于显著性水平 α 的单尾检验，下尾检验的临界值 c 满足 P(X ≤ c) ≤ α，上尾检验则满足 P(X ≥ c) ≤ α。
For a two-tailed test, both tails must be examined. The critical values are the smallest c₁ and largest c₂ such that P(X ≤ c₁) ≤ α/2 and P(X ≥ c₂) ≤ α/2. The CDF table directly provides the lower tail; the upper tail is obtained by complementation.
对于双尾检验，必须检查两个尾部。临界值为最小的 c₁ 和最大的 c₂，使得 P(X ≤ c₁) ≤ α/2 且 P(X ≥ c₂) ≤ α/2。CDF 表直接提供下尾概率；上尾概率通过互补获得。
11. Common Errors and Exam Tips | 常见错误与考试建议
Several common pitfalls appear frequently in examinations. First, students often confuse P(X < x) with P(X ≤ x). Remember that P(X < x) = P(X ≤ x − 1) for integer-valued distributions. Second, when using complementary probabilities, double-check which inequality is required before applying 1 − F(x).
几个常见陷阱在考试中频繁出现。第一，学生经常混淆 P(X < x) 与 P(X ≤ x)。记住对于整数取值的分布，P(X < x) = P(X ≤ x − 1)。第二，使用互补概率时，在应用 1 − F(x) 之前仔细确认所需的不等式方向。
- Always state the distribution clearly: X ~ B(n, p) before calculation.
- 在计算前明确写出分布：X ~ B(n, p)。
- If using statistical tables, check the correct n row and p column.
- 如果使用统计表，检查正确的 n 行和 p 列。
- For higher-tier questions where n > 20, you may need to use a normal approximation with continuity correction — this is a further topic covered separately.
- 对于 n > 20 的高阶题目，可能需要使用带有连续性校正的正态近似——这是另外单独讲解的进阶主题。
12. Summary | 总结
The cumulative distribution function of the binomial distribution is a powerful tool that consolidates individual probabilities into a single cumulative value. By mastering the definition F(x) = P(X ≤ x), the complementary relationship P(X > x) = 1 − F(x), and interval calculations F(b) − F(a − 1), you can approach a wide range of exam problems with confidence.
二项分布的累积分布函数是一个强大的工具，它将各个单独的概率整合为单一的累积值。通过掌握定义 F(x) = P(X ≤ x)、互补关系 P(X > x) = 1 − F(x) 以及区间计算 F(b) − F(a − 1)，您可以自信地应对各种考试问题。
In this article, we have covered the definition, table-reading skills, calculator usage, worked examples, and hypothesis testing applications. Regular practice with past papers will reinforce these skills and build the fluency required for A-Level Further Mathematics success.
本文涵盖了定义、查表技巧、计算器使用、例题和假设检验应用。通过定期练习历年真题，您将巩固这些技能并培养 A-Level 进阶数学成功所需的熟练度。
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