Cumulative Probabilities | 累积概率

📚 Cumulative Probabilities | 累积概率

In A‑Level Mathematics, cumulative probability is the likelihood that a random variable takes a value less than or equal to a given point. It forms the backbone of statistical inference, from hypothesis testing to confidence intervals, and appears across both discrete and continuous distributions. Mastering cumulative probabilities means you can use tables efficiently, standardise values for the normal distribution, and relate P(X ≤ x) to non‑cumulative forms.

在 A‑Level 数学中，累积概率是指随机变量取值小于或等于某一给定点的可能性。它是假设检验、置信区间等统计推断的基石，涉及离散分布和连续分布。掌握累积概率意味着你能高效地使用表格，对正态分布进行标准化，并建立 P(X ≤ x) 与非累积形式之间的换算关系。

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1. What are Cumulative Probabilities? | 什么是累积概率？

A cumulative probability is the sum of probabilities for all outcomes up to and including a specific value. For a discrete random variable X, the cumulative probability at x is written P(X ≤ x) = Σ P(X = k) for all k ≤ x. For continuous variables, it is the area under the probability density curve from −∞ to x.

累积概率是指所有小于等于某一特定值的所有结果概率之和。对于离散型随机变量 X，x 处的累积概率记为 P(X ≤ x) = Σ P(X = k)，其中 k 取遍所有 ≤ x 的值。对于连续型变量，它是概率密度曲线下从 −∞ 到 x 的面积。

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2. Notation: P(X ≤ x) | 符号表示：P(X ≤ x)

Edexcel exam questions routinely use the notation P(X ≤ x), P(X < x), P(X ≥ x) and P(X > x). The key is to handle strict and non‑strict inequalities correctly. For discrete distributions, P(X < x) = P(X ≤ x−1), while for continuous distributions P(X < x) = P(X ≤ x) because the probability of exactly x is zero.

Edexcel 考试中常用 P(X ≤ x)、P(X < x)、P(X ≥ x) 和 P(X > x) 等符号。关键在于正确处理严格与不严格的不等号。对于离散分布，P(X < x) = P(X ≤ x−1)；而对于连续分布，由于单点概率为零，P(X < x) = P(X ≤ x)。

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3. Cumulative Probabilities for Discrete Distributions | 离散分布的累积概率

In discrete cases, cumulative probabilities are obtained by adding the exact probabilities one by one. For a binomial distribution B(n, p), you can build the cumulative table from the probability mass function P(X = k) = ⁿCₖ pᵏ(1−p)ⁿ⁻ᵏ. In practice, you use pre‑computed tables to save time.

离散情形下，累积概率通过逐一累加精确概率得到。对于二项分布 B(n, p)，可由概率质量函数 P(X = k) = ⁿCₖ pᵏ(1−p)ⁿ⁻ᵏ 建立累积表。实际操作中，我们使用预先计算好的表格以节省时间。

For example, if X ~ B(5, 0.3), the individual probabilities are: P(0)=0.16807, P(1)=0.36015, P(2)=0.30870, P(3)=0.13230, P(4)=0.02835, P(5)=0.00243. The cumulative probabilities become P(X ≤ 0)=0.16807, P(X ≤ 1)=0.52822, P(X ≤ 2)=0.83692, P(X ≤ 3)=0.96922, P(X ≤ 4)=0.99757, P(X ≤ 5)=1.

例如，若 X ~ B(5, 0.3)，其单项概率为：P(0)=0.16807，P(1)=0.36015，P(2)=0.30870，P(3)=0.13230，P(4)=0.02835，P(5)=0.00243。对应的累积概率为 P(X ≤ 0)=0.16807，P(X ≤ 1)=0.52822，P(X ≤ 2)=0.83692，P(X ≤ 3)=0.96922，P(X ≤ 4)=0.99757，P(X ≤ 5)=1。

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4. Using Binomial Cumulative Distribution Tables | 使用二项分布累积分布表

Edexcel provides cumulative binomial tables for various n and p. A typical excerpt for n=10 is shown below. The value at the intersection of x=3 and p=0.25 is 0.7759, meaning P(X ≤ 3)=0.7759 when X ~ B(10, 0.25).

Edexcel 提供不同 n 与 p 下的二项累积分布表。以下为 n=10 的片段示例。当 x=3 与 p=0.25 交叉处的值为 0.7759，表示若 X ~ B(10, 0.25)，则 P(X ≤ 3)=0.7759。

x	p=0.10	p=0.25	p=0.50
0	0.3487	0.0563	0.0010
1	0.7361	0.2440	0.0107
2	0.9298	0.5256	0.0547
3	0.9872	0.7759	0.1719

To find P(X ≥ 4) when X ~ B(10, 0.25), remember P(X ≥ 4) = 1 − P(X ≤ 3). From the table, P(X ≤ 3)=0.7759, so P(X ≥ 4)=1 − 0.7759 = 0.2241.

当 X ~ B(10, 0.25) 时求 P(X ≥ 4)，牢记 P(X ≥ 4) = 1 − P(X ≤ 3)。查表得 P(X ≤ 3)=0.7759，因此 P(X ≥ 4)=1 − 0.7759 = 0.2241。

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5. Example: Binomial Cumulative Probability Calculation | 示例：二项累积概率计算

Question: A biased coin lands on heads with probability 0.4. The coin is tossed 15 times. Find the probability of getting at most 7 heads.

题目：一枚偏置硬币出现正面的概率为 0.4。抛掷该硬币 15 次。求得到至多 7 次正面的概率。

Let X ~ B(15, 0.4). We need P(X ≤ 7). From cumulative binomial tables for n=15, p=0.4, the entry for x=7 is 0.7869 (table value may vary slightly depending on rounding). Therefore, the probability is 0.7869.

设 X ~ B(15, 0.4)。我们需要 P(X ≤ 7)。查阅 n=15, p=0.4 的二项累积表，x=7 对应的值为 0.7869（表格数值因舍入可能略有差异）。因此概率为 0.7869。

If the question asked for P(5 ≤ X ≤ 9), compute P(X ≤ 9) − P(X ≤ 4). Using the table, P(X ≤ 9)=0.9662, P(X ≤ 4)=0.2173, giving 0.9662 − 0.2173 = 0.7489.

如果题目要求 P(5 ≤ X ≤ 9)，则计算 P(X ≤ 9) − P(X ≤ 4)。查表得 P(X ≤ 9)=0.9662，P(X ≤ 4)=0.2173，结果为 0.9662 − 0.2173 = 0.7489。

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6. Cumulative Distribution Function (CDF) | 累积分布函数 (CDF)

The cumulative distribution function F(x) gives P(X ≤ x) for all real numbers x. For discrete variables, F(x) is a step function that jumps at each possible value. For continuous variables, F(x) is a smooth, non‑decreasing function ranging from 0 to 1.

累积分布函数 F(x) 对任意实数 x 给出 P(X ≤ x)。对于离散变量，F(x) 是一个阶梯函数，在每个可能的取值处跳跃。对于连续变量，F(x) 是光滑的、非递减的函数，取值范围从 0 到 1。

Understanding F(x) is essential for linking probability density functions (pdf) and cumulative probabilities: for a continuous random variable with density f(t), F(x) = ∫₋∞ˣ f(t) dt.

理解 F(x) 对于将概率密度函数 (pdf) 与累积概率联系起来至关重要：对于具有密度 f(t) 的连续随机变量，F(x) = ∫₋∞ˣ f(t) dt。

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7. Cumulative Probabilities for Continuous Distributions | 连续分布的累积概率

For a continuous distribution, you cannot list P(X = x) because it is always 0. Instead, probabilities are areas under the curve, and cumulative probability P(X ≤ c) is the area to the left of c. The normal distribution N(μ, σ²) is the most important continuous distribution at A‑Level.

对于连续分布，你无法列出 P(X = x)，因为它恒为 0。相反，概率是曲线下的面积，累积概率 P(X ≤ c) 是 c 左侧的面积。正态分布 N(μ, σ²) 是 A‑Level 中最重要的连续分布。

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8. The Normal Cumulative Distribution Function Φ(z) | 正态累积分布函数 Φ(z)

The standard normal distribution Z ~ N(0, 1) has cumulative function Φ(z) = P(Z ≤ z). Tables give Φ(z) for positive z to 3 or 4 decimal places. Because the curve is symmetric, Φ(−z) = 1 − Φ(z). Edexcel’s formula booklet provides the standard normal cumulative distribution table.

标准正态分布 Z ~ N(0, 1) 的累积函数为 Φ(z) = P(Z ≤ z)。表格通常给出正数 z 对应的 Φ(z) 至三位或四位小数。由于曲线对称，Φ(−z) = 1 − Φ(z)。Edexcel 的公式手册提供了标准正态累积分布表。

z	Φ(z)	z	Φ(z)
0.00	0.5000	1.00	0.8413
0.50	0.6915	1.50	0.9332
1.00	0.8413	2.00	0.9772
1.96	0.9750	2.50	0.9938

For z = 1.96, Φ(1.96) = 0.9750, meaning 97.5% of the standard normal distribution lies to the left of 1.96. This is the foundation of 95% confidence intervals.

对于 z = 1.96，Φ(1.96) = 0.9750，意味着标准正态分布下 97.5% 的面积位于 1.96 左侧。这是 95% 置信区间的基础。

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9. Standardising to Use Normal Tables | 标准化以使用正态表

If X ~ N(μ, σ²), convert to the standard normal variable Z = (X − μ)/σ. Then P(X ≤ x) = P(Z ≤ (x−μ)/σ) = Φ((x−μ)/σ). This process is called standardisation and is tested in virtually every normal distribution problem.

若 X ~ N(μ, σ²)，可转化为标准正态变量 Z = (X − μ)/σ。那么 P(X ≤ x) = P(Z ≤ (x−μ)/σ) = Φ((x−μ)/σ)。这个过程称为标准化，几乎每个正态分布问题都会考到。

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10. Example: Normal Cumulative Probability Calculation | 示例：正态累积概率计算

Question: The mass of apples from an orchard is normally distributed with mean 150 g and standard deviation 20 g. Find the proportion of apples weighing less than 175 g.

题目：某果园苹果的质量服从正态分布，均值为 150 g，标准差为 20 g。求质量小于 175 g 的苹果所占比例。

Let M ~ N(150, 20²). Then P(M < 175) = P(Z < (175−150)/20) = P(Z < 1.25). From tables, Φ(1.25) ≈ 0.8944. So about 89.44% of apples weigh less than 175 g.

设 M ~ N(150, 20²)。则 P(M < 175) = P(Z < (175−150)/20) = P(Z < 1.25)。查表 Φ(1.25) ≈ 0.8944。因此约 89.44% 的苹果质量小于 175 g。

For a weight exceeding 130 g: P(M > 130) = 1 − P(Z < (130−150)/20) = 1 − P(Z < −1.00) = 1 − (1 − Φ(1.00)) = Φ(1.00) = 0.8413. Hence 84.13% of apples weigh more than 130 g.

对于质量超过 130 g 的情况：P(M > 130) = 1 − P(Z < (130−150)/20) = 1 − P(Z < −1.00) = 1 − (1 − Φ(1.00)) = Φ(1.00) = 0.8413。因此 84.13% 的苹果质量超过 130 g。

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11. Relationship Between Cumulative and Non‑Cumulative Probabilities | 累积与非累积概率的关系

In many problems you are given cumulative probabilities and must extract the probability of an exact value or an interval. For discrete variables: P(X = k) = P(X ≤ k) − P(X ≤ k−1). For continuous variables, interval probabilities are P(a < X < b) = P(X ≤ b) − P(X ≤ a).

在许多问题中，你会得到累积概率，需要求出精确值或区间的概率。对于离散变量：P(X = k) = P(X ≤ k) − P(X ≤ k−1)。对于连续变量，区间概率为 P(a < X < b) = P(X ≤ b) − P(X ≤ a)。

This simple subtraction is the key to moving between the cumulative distribution function and the original probability distribution, whether it is binomial, Poisson, or normal.

这个简单的减法是在累积分布函数与原始概率分布之间转换的关键，无论是二项分布、泊松分布还是正态分布。

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12. Common Mistakes and Tips | 常见错误与提示

1. Confusing strict and non‑strict inequalities: In discrete distributions, remember to subtract 1 when converting P(X < x) to P(X ≤ x−1). In continuous distributions, the equality makes no difference.

1. 混淆严格与非严格不等号：离散分布中，将 P(X < x) 转化为 P(X ≤ x−1) 时记得减 1。连续分布中，等号没有影响。

2. Reading the wrong table column: Always double‑check you are using the correct p and n for binomial, or the correct tail for normal (most Edexcel tables give cumulative lower tail Φ(z)).

2. 读错表格列：务必核实二项分布使用了正确的 p 和 n，正态分布使用了正确的尾部（Edexcel 大部分表格给出下侧累积 Φ(z)）。

3. Forgetting to standardise: Never plug a non‑standard normal value directly into Φ(z) tables without computing the z‑score first.

3. 忘记标准化：切勿将非标准正态值直接代入 Φ(z) 表，必须先计算 z 分数。

4. Misapplying the 1 − rule: P(X ≥ k) = 1 − P(X ≤ k−1) for discrete, and P(X > c) = 1 − Φ(c) for standard normal. Adjust accordingly.

4. 错误使用 1 − 规则：离散时 P(X ≥ k) = 1 − P(X ≤ k−1)，标准正态时 P(Z > c) = 1 − Φ(c)。需相应调整。

Practise with past exam papers, and always sketch a quick diagram for normal problems to visualise the area you need.

通过历年真题练习，并在求解正态问题时随手画出简图，以直观判断所需面积。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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